Whereas the @ specification of the OP's MWE seems to be locked into stone (can't be changed on the fly), the same is not true of the < specification, which is re-evaluated on the fly.
\documentclass{article}
\usepackage{array}
\newcommand\mycolsep{\emph{ -- }}
\newcommand\myspan{\let\mycolsep\relax\span}
\begin{document}
\begin{tabular}{r<{\mycolsep}@{}l | p{11cm}}
\emph{Jun 2013} & \emph{Aug 2013} & Work position at Company\\
\emph{Aug 2012} \myspan & Other Work position at Firm\\
\emph{Jun 2011} & \emph{Aug 2011} & Work position at Company\\
\emph{Aug 2010} \myspan & Other Work position at Firm\\
\end{tabular}
\end{document}

If you want the result left-aligned, and in light of a comment by the OP, I take a slightly altered approach:
\documentclass{article}
\usepackage{array}
\newcommand\mycolsep{\emph{ -- }}
\newcommand\myspan{\def\mycolsep{\phantom{\emph{ -- }}}&}
\begin{document}
\begin{tabular}{r<{\mycolsep}@{}l | p{11cm}}
\emph{Jun 2013} & \emph{Aug 2013} & Work position at Company\\
\emph{Aug 2012} \myspan & Other Work position at Firm\\
\emph{Jun 2011} & \emph{Aug 2011} & Work position at Company\\
\emph{Jul 2010} \myspan & Other Work position at Firm\\
\end{tabular}
\end{document}

\spanin some of the answers here on StackExchange. And on snippets vs complete examples: I just thought the snippets would be enough to show you what I would like to complete. – tukusejssirs Mar 13 '19 at 16:50\span. … Next time, I include the complete striped-down examples. :) – tukusejssirs Mar 13 '19 at 17:25