Two questions about typesetting a Roman missal

I have updated the longtable solution, using the class scrartcl from the KOMA-script bundle. I have also added new font, Palantino, a very nice and readable font.

Title is set using KOMA-script features. Necessary babel environments are piped into the column.

You have now a better paracol-solution than before, but I am still of the opinion that longtable is a better and safer environment for typesetting parallel column, unless you have very long paragraphs which may be a challenge for good page breaks.

Working paracol MWE is at the end of the answer:

longtable

\documentclass[table, latin, slovene, fontsize=12pt, parskip=full,titlepage=false]{scrartcl}
\usepackage{tgpagella}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb, array, longtable}
\usepackage{xcolor, babel}
\usepackage[babel=true]{microtype}

\usepackage[layout=a5paper,vmargin=0.4in, tmargin=0.6in,%
                      hmargin=0.4in,paperheight=216mm,%
                      includehead, footskip=1.5\baselineskip,%
                      paperwidth=154mm]{geometry}


\newcommand{\+}{{\maltese}}
\newcolumntype{L}{>{\begin{otherlanguage}{latin}}p{\dimexpr(0.5\linewidth-\tabcolsep-0.5\arrayrulewidth)}<{\end{otherlanguage}\vspace{0.5\baselineskip}}}
\newcolumntype{S}{>{\begin{otherlanguage}{slovene}}p{\dimexpr(0.5\linewidth-\tabcolsep-0.5\arrayrulewidth)}<{\end{otherlanguage}\vspace{0.5\baselineskip}}}

\setlength{\tabcolsep}{1em}
\arrayrulecolor{gray}
\renewcommand*\titlepagestyle{empty}
\renewcommand\pagestyle{scrheadings}
\setkomafont{subject}{\LARGE\bfseries}
\setkomafont{title}{\large\mdseries}
\setkomafont{subtitle}{\color{red}\footnotesize}

\begin{document}

\subject{Red svete maše}
\title{\large Pristopne molitve (KLEČIMO)}
\subtitle{Mašnik pripravi kelih, odpre knjigo, nato gre pred stopnice, se pokriža in moli. \mbox{Z njim} se pokrižam, pokleknem in molim (po tiho): }
\date{\vspace{-4\baselineskip}}

\maketitle

\begin{longtable}[c]{@{}L|S@{}}
Mašnik: In nomine \+ Patris, et  Fílii, et Spíritus Sancti. Amen. &
V imenu \+ Očeta in Sina in Svetega Duha. Amen.\\

Mašnik: (Ant.) Introíbo ad altáre Dei. &
Stopil bom k Božjemu oltarju.\\

\bfseries Strežnik: Ad Deum qui lætíficat iuventútem meam. &
K Bogu, ki razveseljuje mojo mladost.\\

M. Iúdica me, Deus, et discérne cáusam meam de gente non sancta: ab hómine iníquo et dolóso érue me.&
Sodi me, Bog, in razsodi mojo pravdo zoper nezvesto ljudstvo; krivičnega in zvijačnega človeka me reši.\\

\bfseries S. Quia tu es, Deus, fortitúdo mea: quare me repulísti, et quare tristis incédo, dum afflígit me inimícus?&
Ker si ti moja moč, o Bog, zakaj si me zavrgel? In zakaj hodim žalosten, ko me stiska sovražnik? \\

M. Emítte lucem tuam et veritátem tuam: ipsa me deduxérunt et adduxérunt in montem sanctum tuum, et in tabernácula tua.&
Pošlji svojo luč in svojo resnico, da me popeljeta na tvojo sveto goro in v tvoje šotore.\\

\bfseries S. Et introíbo ad altáre Dei: ad Deum qui lætíficat iuventútem meam. &
In stopil bom k Božjemu oltarju; k Bogu, ki razveseljuje mojo mladost.\\

M. Confitébor tibi in cíthara, Deus, Deus meus: quare tristis es anima mea, et quare contúrbas me? &
Hvalil te bom s citrami, Bog, moj Bog! Zakaj si žalostna, moja duša, in zakaj me vznemirjaš?\\

\bfseries S. Spera in Deo, quóniam adhuc confitébor illi: salutáre vultus mei, et Deus meus.&
Upaj v Boga, zakaj še ga bom hvalil; on je pomoč mojega obličja in moj Bog.\\\end{longtable}

\end{document}paracol

\documentclass[a6paper,pagesize,12pt]{book}
\usepackage{lmodern}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb, array, paracol}
\usepackage{xcolor}
\usepackage[latin,slovene]{babel}        % Thanks to David Purton
\usepackage[babel=true]{microtype}       % Important for nice typesetting

\usepackage{showframe}
\renewcommand*\ShowFrameColor{\color{red}}
\renewcommand*\ShowFrameLinethickness{.4pt}

\usepackage[layout=a5paper,vmargin=0.4in,tmargin=0.6in,hmargin=0.4in,paperheight=216mm,paperwidth=154mm,showframe=false]{geometry}
\pagenumbering{gobble}

\usepackage[parfill]{parskip}
\setlength\parindent{0pt}

\newcommand\kom[1]{{\vspace{0.2cm}\begin{center}\footnotesize \color{red} #1 \end{center}}}
\newcommand{\+}{{\maltese}}

\newcolumntype{N}{p{\dimexpr(0.5\linewidth-\tabcolsep-0.5\arrayrulewidth)}}

\begin{document}

\begin{center}
{\LARGE \textbf{\textsc{Red svete maše}}} \\ \vspace{1cm}
\textbf{Pristopne molitve (KLEČIMO)} \\
\kom{Mašnik pripravi kelih, odpre knjigo, nato gre pred stopnice, se pokriža in moli. Z njim se pokrižam, pokleknem in molim (po tiho):}
\end{center}


    \begin{paracol}{2}
      \switchcolumn[1]* Text in language 1, which is much longer than you thin
      \switchcolumn[0]  Text in language 1, which is much longer than you thin
     \switchcolumn[1]* Text in language 1
      \switchcolumn[0]  Text in language 2
     \switchcolumn[1]* Text in language 1
      \switchcolumn[0]  Text in language 2

    \end{paracol}

\end{document}

As requested, here’s a solution with paracol, a newer and more flexible package for parallel columns that is still being updated. This is closer to what you wanted, and will work better than my original answer if you want the columns to have different widths or spacing.

As you can see, I defined some macros for convenience. It uses Unicode and modern fonts, and should be compiled in LuaLaTeX, but it would not be difficult to adapt to PDFTeX if you really wanted to.

I also added a couple of packages: babel to support multilingual text (such as hyphenation patterns and font switching if necessary) and microtype to support font expansion, which should reduce the need for sloppiness.

\documentclass[a6paper,pagesize,12pt]{book}
\usepackage[layout=a5paper, vmargin=0.4in, tmargin=0.6in, hmargin=0.4in,
            paperheight=216mm, paperwidth=154mm, showframe=false
           ]{geometry}
\usepackage[svgnames]{xcolor}
\usepackage[latin, main=slovene]{babel}
\usepackage{fontspec, microtype, parskip, paracol}

\defaultfontfeatures{Scale=MatchUppercase}
\babelfont{rm}
          [Scale=1.0,
           Language=Default,
           Ligatures={Common, TeX}
          ]{TeX Gyre Schola} % Or your font of choice.

\newfontfamily\symbolfont{DejaVu Sans}

\pagenumbering{gobble}
\setlength\columnseprule{0.4pt}
\sloppy

\newcommand\kom[1]{{\vspace{0.2cm}\begin{center}\footnotesize \color{red} #1 \end{center}}}
\newcommand{\+}{{\symbolfont\symbol{"2720}}} % Maltese cross.

\newcommand\Masnik{\foreignlanguage{slovene}{Mašnik}}
\newcommand\Streznik{\foreignlanguage{slovene}{Strežnik}}

\newcommand\chunks[2]{%
\begin{leftcolumn*}\begin{otherlanguage}{latin}%
{#1}%
\end{otherlanguage}\end{leftcolumn*}%
\begin{rightcolumn}\begin{otherlanguage}{slovene}%
{#2}%
\end{otherlanguage}\end{rightcolumn}%
}

\begin{document}

\begin{center}
{\LARGE \textbf{\textsc{Red svete maše}}} \\ \vspace{1cm}
\textbf{Pristopne molitve (KLEČIMO)} \\
\kom{Mašnik pripravi kelih, odpre knjigo, nato gre pred stopnice, se pokriža in moli. Z njim se pokrižam, pokleknem in molim (po tiho):}
\end{center}

\begin{paracol}{2}
  \chunks{\Masnik: In nomine \+ Patris, et Filii et Spiritus Sancti. Amen.}
         {V imenu \+ Očeta in Sina in Svetega Duha. Amen.}
  \chunks{\Masnik: (Ant.) Introibo ad altare Dei.}
         {Stopil bom k Božjemu oltarju.}
  \chunks{\textbf{\Streznik: Ad Deum qui lætíficat iuventútem meam.}}
         {\textbf{K Bogu, ki razveseljuje mojo mladost.}}
  \chunks{M. Iúdica me, Deus, et discérne cáusam meam de gente non sancta: ab hómine iníquo et dolóso érue me.}
         {Sodi me, Bog, in razsodi mojo pravdo zoper nezvesto ljudstvo; krivičnega in zvijačnega človeka me reši.}
  \chunks{\textbf{S. Quia tu es, Deus, fortitúdo mea: quare me repulísti, et quare tristis incédo, dum afflígit me inimícus?}}
         {\textbf{Ker si ti moja moč, o Bog, zakaj si me zavrgel? In zakaj hodim žalosten, ko me stiska sovražnik?}}
  \chunks{M. Emítte lucem tuam et veritátem tuam: ipsa me deduxérunt et adduxérunt in montem sanctum tuum, et in tabernácula tua.}
         {Pošlji svojo luč in svojo resnico, da me popeljeta na tvojo sveto goro in v tvoje šotore.}
  \chunks{\textbf{S. Et introíbo ad altáre Dei: ad Deum qui lætíficat iuventútem meam.}}
         {\textbf{In stopil bom k Božjemu oltarju; k Bogu, ki razveseljuje mojo mladost.}}
  \chunks{M. Confitébor tibi in cíthara, Deus, Deus meus: quare tristis es anima mea, et quare contúrbas me?}
         {Hvalil te bom s citrami, Bog, moj Bog! Zakaj si žalostna, moja duša, in zakaj me vznemirjaš?}
\end{paracol}
\end{document}

Previous Answer

Here is a version that uses the pdfcolparcolumns update to parcolumns. It should at least work with color.

\documentclass[a6paper,pagesize,12pt]{book}
\usepackage[svgnames]{xcolor}
\usepackage[latin, main=slovene]{babel}
\usepackage{fontspec}
\usepackage{microtype}
\usepackage[layout=a5paper, vmargin=0.4in, tmargin=0.6in, hmargin=0.4in,
            paperheight=216mm, paperwidth=154mm, showframe=false
           ]{geometry}

\usepackage{pdfcolparcolumns}
\usepackage{parskip}

\defaultfontfeatures{Scale=MatchUppercase}
\babelfont{rm}
          [Scale=1.0,
           Language=Default,
           Ligatures={Common, TeX}
          ]{TeX Gyre Schola} % Or your font of choice.

\newfontfamily\symbolfont{DejaVu Sans}

\pagenumbering{gobble}

\newcommand\kom[1]{{\vspace{0.2cm}\begin{center}\footnotesize \color{red} #1 \end{center}}}
\newcommand{\+}{{\symbolfont\symbol{"2720}}} % Maltese cross.

\newcommand\Masnik{\foreignlanguage{slovene}{Mašnik}}
\newcommand\Streznik{\foreignlanguage{slovene}{Strežnik}}

\newcommand\chunks[2]{%
\colchunk[1]{\begin{otherlanguage}{latin}\par{#1}\end{otherlanguage}}%
\colchunk[2]{\begin{otherlanguage}{slovene}\par{#2}\end{otherlanguage}}%
\colplacechunks%
}

\begin{document}

\begin{center}
{\LARGE \textbf{\textsc{Red svete maše}}} \\ \vspace{1cm}
\textbf{Pristopne molitve (KLEČIMO)} \\
\kom{Mašnik pripravi kelih, odpre knjigo, nato gre pred stopnice, se pokriža in moli. Z njim se pokrižam, pokleknem in molim (po tiho):}
\end{center}

\begin{parcolumns}[distance=1em, nofirstindent, rulebetween, sloppy]{2}

  \chunks{\Masnik: In nomine \+ Patris, et Filii et Spiritus Sancti. Amen.}
         {V imenu \+ Očeta in Sina in Svetega Duha. Amen.}

  \chunks{\Masnik: (Ant.) Introibo ad altare Dei.}
         {Stopil bom k Božjemu oltarju.}

  \chunks{\textbf{\Streznik: Ad Deum qui lætíficat iuventútem meam.}}
         {\textbf{K Bogu, ki razveseljuje mojo mladost.}}

  \chunks{M. Iúdica me, Deus, et discérne cáusam meam de gente non sancta: ab hómine iníquo et dolóso érue me.}
         {Sodi me, Bog, in razsodi mojo pravdo zoper nezvesto ljudstvo; krivičnega in zvijačnega človeka me reši.}

  \chunks{\textbf{S. Quia tu es, Deus, fortitúdo mea: quare me repulísti, et quare tristis incédo, dum afflígit me inimícus?}}
         {\textbf{Ker si ti moja moč, o Bog, zakaj si me zavrgel? In zakaj hodim žalosten, ko me stiska sovražnik?}}

  \chunks{M. Emítte lucem tuam et veritátem tuam: ipsa me deduxérunt et adduxérunt in montem sanctum tuum, et in tabernácula tua.}
         {Pošlji svojo luč in svojo resnico, da me popeljeta na tvojo sveto goro in v tvoje šotore.}

  \chunks{\textbf{S. Et introíbo ad altáre Dei: ad Deum qui lætíficat iuventútem meam.}}
         {\textbf{In stopil bom k Božjemu oltarju; k Bogu, ki razveseljuje mojo mladost.}}

  \chunks{M. Confitébor tibi in cíthara, Deus, Deus meus: quare tristis es anima mea, et quare contúrbas me?}
         {Hvalil te bom s citrami, Bog, moj Bog! Zakaj si žalostna, moja duša, in zakaj me vznemirjaš?}

\end{parcolumns}
\end{document}

I simply copied the text from your MWE, and do not speak Slovene, so I apologize for any errors.

Edit: Thanks to Sveinung for pointing out an error with the spacing. It explicitly starts a new paragraph with every chunk now. To change the paragraph spacing, \usepackage[skip=\someLength]{parskip}.

I've found a solution using paracol. The basic idea is to synchronize columns with switching from leftcolumn* to rightcolumn. This code puts every next left paragraph at the bottom of every previous right paragraph.

\documentclass[a6paper,pagesize,12pt]{book}

\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb}
\usepackage{color}

\usepackage[layout=a5paper,vmargin=0.4in,tmargin=0.6in,hmargin=0.4in,paperheight=216mm,paperwidth=154mm,showframe=false]{geometry}

\usepackage{paracol}

\pagenumbering{gobble}

\usepackage[parfill]{parskip}
\setlength\parindent{0pt}

\newcommand\kom[1]{{\vspace{0.2cm}\begin{center}\footnotesize \color{red} #1 \end{center} \vspace{0.2cm}}}
\newcommand{\+}{{\maltese}}

\sloppy

\begin{document}

\begin{center}
{\LARGE \textbf{\textsc{Red svete maše}}} \\ \vspace{0.8cm}
\textbf{Pristopne molitve (KLEČIMO)} \\
\kom{Mašnik pripravi kelih, odpre knjigo, nato gre pred stopnice, se pokriža in moli. Z njim se pokrižam, pokleknem in molim (po tiho):}
\end{center}

\begin{paracol}{2}

\setlength\columnseprule{0.4pt}

\begin{leftcolumn*}
Mašnik: In nomine \+ Patris, et  Fílii, et Spíritus Sancti. Amen.
\end{leftcolumn*}

\begin{rightcolumn}
V imenu \+ Očeta in Sina in Svetega Duha. Amen.
\end{rightcolumn}

\begin{leftcolumn*}
Mašnik: (Ant.) Introíbo ad altáre Dei.
\end{leftcolumn*}

\begin{rightcolumn}
Stopil bom k Božjemu oltarju.
\end{rightcolumn}

\begin{leftcolumn*}
\textbf{Strežnik: Ad Deum qui lætíficat iuventútem meam.}
\end{leftcolumn*}

\begin{rightcolumn}
K Bogu, ki razveseljuje mojo mladost.
\end{rightcolumn}

\begin{leftcolumn*}
M. Iúdica me, Deus, et discérne cáusam meam de gente non sancta: ab hómine iníquo et dolóso érue me.
\end{leftcolumn*}

\begin{rightcolumn}
Sodi me, Bog, in razsodi mojo pravdo zoper nezvesto ljudstvo; krivičnega in zvijačnega človeka me reši.
\end{rightcolumn}

\begin{leftcolumn*}
\textbf{S. Quia tu es, Deus, fortitúdo mea: quare me repulísti, et quare tristis incédo, dum afflígit me inimícus?}
\end{leftcolumn*}

\begin{rightcolumn}
Ker si ti moja moč, o Bog, zakaj si me zavrgel? In zakaj hodim žalosten, ko me stiska sovražnik?
\end{rightcolumn}

\begin{leftcolumn*}
M. Emítte lucem tuam et veritátem tuam: ipsa me deduxérunt et adduxérunt in montem sanctum tuum, et in tabernácula tua.
\end{leftcolumn*}

\begin{rightcolumn}
Pošlji svojo luč in svojo resnico, da me popeljeta na tvojo sveto goro in v tvoje šotore.
\end{rightcolumn}

\begin{leftcolumn*}
\textbf{S. Et introíbo ad altáre Dei: ad Deum qui lætíficat iuventútem meam.}
\end{leftcolumn*}

\begin{rightcolumn}
In stopil bom k Božjemu oltarju; k Bogu, ki razveseljuje mojo mladost.
\end{rightcolumn}

\begin{leftcolumn*}
M. Confitébor tibi in cíthara, Deus, Deus meus: quare tristis es anima mea, et quare contúrbas me?
\end{leftcolumn*}

\begin{rightcolumn}
Hvalil te bom s citrami, Bog, moj Bog! Zakaj si žalostna, moja duša, in zakaj me vznemirjaš?
\end{rightcolumn}

\begin{leftcolumn*}
\textbf{S. Spera in Deo, quóniam adhuc confitébor illi: salutáre vultus mei, et Deus meus.}
\end{leftcolumn*}

\begin{rightcolumn}
Upaj v Boga, zakaj še ga bom hvalil; on je pomoč mojega obličja in moj Bog.
\end{rightcolumn}

\end{paracol}

\end{document}

The first page of the output looks like this.