  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. In this note we mimic
this theory with finite extensions of free groups, with an emphasis on free
products of the cyclic group $C_p$, for $p$ an odd prime. We show that for $n
\leq p$ Magnus' characterization holds for the $n$-fold free product $C_p^{*n}$
within the class of finite-extensions of free groups. Specifically, if $n \leq
p$ and $G$ is a finitely generated, virtually free, residually nilpotent group
having the same lower central series quotients as $C_p^{*n}$, then $G \cong
C_p^{*n}$. We also show that such a characterization does not hold in the class
of finitely generated groups. That is, we construct a rank 2 residually
nilpotent group $G$ that shares all its lower central series quotients with
$\ffp$, but is not $\ffp$.
