  Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the
plane such that any two parallel edges form a simple closed curve with at least
one vertex in its interior and at least one vertex in its exterior. Pach and
T\'oth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing
Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics
Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of
computing series, 1983) by showing that if no two adjacent edges cross and
every pair of nonadjacent edges cross at most once, then the number of edge
crossings in $G$ is at least $\alpha e^3/n^2$, for a suitable constant
$\alpha>0$. The situation turns out to be quite different if nonparallel edges
are allowed to cross any number of times. It is proved that in this case the
number of crossings in $G$ is at least $\alpha e^{2.5}/n^{1.5}$. The order of
magnitude of this bound cannot be improved.
