  A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that
every edge is crossed at most $k$ times. For $k \leq 4$, Pach and T\'oth proved
a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph,
which is tight for $k=1,2$. For $k=3$, the bound of $6n-12$ has been improved
to $\frac{11}{2}n-11$ and has been shown to be optimal up to an additive
constant for simple graphs. In this paper, we prove that the bound of
$\frac{11}{2}n-11$ edges also holds for non-simple $3$-planar graphs that admit
drawings in which non-homotopic parallel edges and self-loops are allowed.
Based on this result, a characterization of \emph{optimal $3$-planar graphs}
(that is, $3$-planar graphs with $n$ vertices and exactly $\frac{11}{2}n-11$
edges) might be possible, as to the best of our knowledge the densest known
simple $3$-planar is not known to be optimal.
