Closer slanted parallel symbol

Raise and scale.

\documentclass{article}
\usepackage{amsmath,graphicx}

\makeatletter
\newcommand{\newparallel}{\mathrel{\mathpalette\new@parallel\relax}}
\newcommand{\new@parallel}[2]{%
  \begingroup
  \sbox\z@{$#1T$}% get the height of an uppercase letter
  \resizebox{!}{\ht\z@}{\raisebox{\depth}{$\m@th#1/\mkern-5mu/$}}%
  \endgroup
}
\makeatother

\begin{document}

$AB\newparallel CD$

$\scriptstyle AB\newparallel CD$

\end{document}

A different implementation with picture mode. One can easily vary the slope by acting on the second argument to \mathpalette.

\documentclass{article}
\usepackage{amsmath,pict2e}

\makeatletter
\newcommand{\newparallel}{\mathrel{\mathpalette\new@parallel{0.3}}}
\newcommand{\new@parallel}[2]{%
  \begingroup
  \settoheight{\unitlength}{$#1T$}% get the height of an uppercase letter
  \sbox\z@{\new@parallel@slash{#1}{#2}}%
  \mkern0.5mu\copy\z@\mkern-0.5mu\copy\z@\mkern0.5mu
  \endgroup
}
\newcommand{\new@parallel@slash}[2]{%
  \begin{picture}(#2,1)
  \roundcap
  \new@parallel@linethickness{#1}
  \Line(0,0)(#2,1)
  \end{picture}%
}
\newcommand{\new@parallel@linethickness}[1]{%
  \linethickness{%
      \ifx#1\displaystyle \fontdimen8\textfont\else
      \ifx#1\textstyle \fontdimen8\textfont\else
      \ifx#1\scriptstyle \fontdimen8\scriptfont\else
      1.1\fontdimen8\scriptscriptfont\fi\fi\fi 3
  }%
}
\makeatother

\begin{document}

$AB\newparallel CD$ ${\newparallel}/$

$\scriptstyle AB\newparallel CD$ $\scriptstyle{\newparallel}/$

$\scriptscriptstyle AB\newparallel CD$ $\scriptscriptstyle{\newparallel}/$

{\Large$AB\newparallel CD$ ${\newparallel}/$}

\end{document}

This is for Computer Modern fonts; for different fonts, \roundcap might be omitted. In the example I show the standard slash to compare the line thickness.