Parallel texts with header on each page?

The parcolumns package might be useful here. A MWE follows- there are many options that can be tweaked (such as colwidths); see the documentation for more examples.

\documentclass{article}
\usepackage{parcolumns}
\usepackage{lipsum}

\begin{document}

\begin{parcolumns}{2}
 \colchunk{\section{1st column}
 \lipsum }

 \colchunk{\section{2nd column}
 \lipsum}
\end{parcolumns}

\end{document}

You might also find the following post helpful:

How do I create separate columns in LaTeX without text flow?

Okay, here's a hack that uses atbegshi to add a header to parcolumns:

\documentclass{article}
\usepackage{parcolumns}
\usepackage{atbegshi}

\newcommand{\pcheaderfmt}[1]{{\large\sf\textbf{#1}}}
\newcommand{\pcheader}{}
\AtBeginShipout{\pcheader}

\begin{document}
\section{Latin and Piglatin}
    \renewcommand{\pcheader}
        {\par\noindent\pcheaderfmt{Latin}\hfill\pcheaderfmt{Piglatin}%
            \vspace{0.5ex}\hrule\vspace{1.5ex}}
    \pcheader
    \begin{parcolumns}{2}
    %
    \colchunk{Latin}
    %
    \colchunk{Piglatin}
    \colplacechunks
    %
    \colchunk{Latin}
    %
    \colchunk{Piglatin}
    %
    \end{parcolumns}
    \renewcommand{\pcheader}{}
\section{English and Pigenglish}
    \renewcommand{\pcheader}
        {\par\noindent\pcheaderfmt{English}\hfill\pcheaderfmt{Pigenglish?}%
            \vspace{0.5ex}\hrule\vspace{1.5ex}}
    \pcheader
    \begin{parcolumns}{2}
    %
    \colchunk{English}
    %
    \colchunk{Pigenglish}
    %
    \end{parcolumns}
    \renewcommand{\pcheader}{}
\end{document}

I'm not sure I fully understand the requirements of your question, but it seems to me that you might want to try the longtable package along with the fancyhdr package.

\documentclass{article}
\usepackage{longtable}
\usepackage{fancyhdr}
\usepackage{lipsum}

\def\chapterName{}
\def\leftName{}
\def\rightName{}
\newcommand*{\Chapter}[3]{%
  \endlongtable%
  \gdef\chapterName{#1}%
  \gdef\leftName{#2}%
  \gdef\rightName{#3}%
  %% ELIMINATE THE || characters to remove the vertical rules
  \longtable[c]{||p{0.45\textwidth}|p{0.45\textwidth}||}
  \multicolumn{2}{c}{\Large\chapterName} \tabularnewline
  \textbf{\leftName} & \textbf{\rightName} \endhead
}
\let\Title=\chead
% \renewcommand{\headrulewidth}{0pt}  %% OPTIONAL

\begin{document}
\pagestyle{fancy}
\Title{A Polyglot Tale}
\begin{longtable}[c]{||p{0.45\textwidth}|p{0.45\textwidth}||}
\Chapter{The taming of Lipsum}{Language A}{Language B}
\lipsum[1] & \lipsum[2] \\
\lipsum[3] & \lipsum[4] \\
\Chapter{The breaking of Lipsum}{Language C}{Language D}
\lipsum[5] & \lipsum[6] \\
\lipsum[7] & \lipsum[8] 
\end{longtable}

\end{document}

This will show "A Polyglot Tale" centered on every page header, then have a running table head of the chapter name, followed by the language names. The one restriction it currently has (baked into longtable) is that no individual row---which here corresponds to a pair of paragraphs---can span pages. If that is your requirement, then longtable can't help you. But if that's acceptable, then there's no limit on how long a longtable can grow...and in any case, this particular solution finishes and restarts the longtable environment at the close of each chapter. (Also, the various rules are entirely optional; they're there to show the extent of the page and the table.)