  A residually nilpotent group is \emph{$k$-parafree} if all of its lower
central series quotients match those of a free group of rank $k$. Magnus proved
that $k$-parafree groups of rank $k$ are themselves free. We mimic this theory
with surface groups playing the role of free groups. Our main result shows that
the analog of Magnus' Theorem is false in this setting.
