  In Path Set Packing, the input is an undirected graph $G$, a collection
$\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to
decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the
parameterized complexity of Path Set Packing with respect to both natural and
structural parameters. We show that the problem is $W[1]$-hard with respect to
vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree
plus solution size. These results answer an open question raised in COCOON
2018. On the positive side, we present an FPT algorithm parameterized by
feedback vertex number plus maximum degree, and present an FPT algorithm
parameterized by treewidth plus maximum degree plus maximum length of a path in
$\calp$. These positive results complement the hardness of Path Set Packing
with respect to any subset of the parameters used in the FPT algorithms. We
also give a $4$-approximation algorithm for maximum path set packing problem
which runs in FPT time when parameterized by feedback edge number.
