In parrun.sty, five internal commands use the \hrule primitive. Patching these commands with the help of the etoolbox package to remove \hrule seems to do the trick.
\documentclass{article}
\usepackage{parrun}
\usepackage{lipsum}
\usepackage{xcolor}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\b@xbalance}{\hrule}{}{}{}
\patchcmd{\v@idb@xbalance}{\hrule}{}{}{}
\patchcmd{\v@idb@xone}{\hrule}{}{}{}
\patchcmd{\v@idb@xtwo}{\hrule}{}{}{}
\patchcmd{\splitb@x}{\hrule}{}{}{}% relevant for the MWE
\makeatother
\begin{document}
\begin{fframe}
\lipsum[1-2]
\end{fframe}
\begin{sframe}
\color{blue}\lipsum[3-4]
\end{sframe}
\Place*
\end{document}

\patchcmdis indeed the crucial one. – Daniel Harbour Sep 26 '12 at 14:45