Mix align and enumerate

Aligning at the equation tags

align* is not required if you just wish to align the items. Here's an example using multienum:

\documentclass{article}
\usepackage{multienum}
\renewcommand{\regularlisti}{\setcounter{multienumi}{0}%
  \renewcommand{\labelenumi}%
  {\addtocounter{multienumi}{1}\alph{multienumi})}}
\begin{document}
\begin{multienumerate}
\mitemxxx{\(x^2 + y^2 = 1\)}{\(a + b = c\)}{\(r-x = y+z\)}
\mitemxxx{\(f - y = z\)}{\(a - b = 2d\)}{\(r+x = 2y-3z\)}
\end{multienumerate}
\end{document}

Aligning relation symbols and equation tags

If you wish to align both at relation symbols and on equation numbers, I recommend to use the alignat* environment. Mark alignment points by &, further use & to separate columns. The argument to alignat* specifies the number of columns. Here we've got six columns: three for the tags and three for the equations.

\documentclass{article}
\usepackage{amsmath}
\newcounter{eqn}
\renewcommand*{\theeqn}{\alph{eqn})}
\newcommand{\num}{\refstepcounter{eqn}\text{\theeqn}\quad}
\begin{document}
\begin{alignat*}{6}
  \num&& x^2 + y^2 &= 1 \qquad& \num&& a + b &= c
          &  \num&& r-x &= y+z \\
  \num&\quad& f - y &= z      & \num&& a - b &= 2d
    \qquad&  \num&& r+x &= 2y-3z
\end{alignat*}
\end{document}

I used automatic numbering. You could do it manually as well, just take care of the font, the tags should be in text font. So they don't look like variables.