With the following code...
\documentclass[draft,10pt,a5paper]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{fontspec}
\setmainfont{TeX Gyre Pagella}
\usepackage{parallel}
\setlength\parindent{0pt}
\begin{document}
\section{My Section} % kill this line and the result is OK
BAD: % or kill this line and the result is ok
\begin{Parallel}{.47\textwidth}{.47\textwidth}
\ParallelLText{Lorem ipsum dolor sit amet.}
\ParallelRText{Sed do eiusmod tempor incididunt ut labore et dolore.}
\end{Parallel}
\bigskip
GOOD:
\begin{Parallel}{.47\textwidth}{.47\textwidth} % same as above...
\ParallelLText{Lorem ipsum dolor sit amet.}
\ParallelRText{Sed do eiusmod tempor incididunt ut labore et dolore.}
\end{Parallel}
\end{document}
I get this result:

See the blank line between "BAD" and "Lorem ipsum"?
To me, it seems like a parallel bug. I know there are other package for typesetting parallel text, such as paracol or parcolumns, but usually I have better results with parallel (fewer widows for instance).
~\par\vskip -\baselineskipbefore theBADline provides a workaround, but not really a fix. – Steven B. Segletes Jun 03 '13 at 13:09\everyparset by\@afterheadingand so\clubpenaltyis not reset correctly after the first paragraph ("BAD"). You could add\makeatletter\clubpenalty \@clubpenalty\makeatotherbefore\begin{parallel}. – Ulrike Fischer Jun 03 '13 at 13:14