  Given a graph $G = (V,E)$, $A \subseteq V$, and integers $k$ and $\ell$, the
\textsc{$(A,\ell)$-Path Packing} problem asks to find $k$ vertex-disjoint paths
of length $\ell$ that have endpoints in $A$ and internal points in $V \setminus
A$. We study the parameterized complexity of this problem with parameters
$|A|$, $\ell$, $k$, treewidth, pathwidth, and their combinations. We present
sharp complexity contrasts with respect to these parameters. Among other
results, we show that the problem is polynomial-time solvable when $\ell \le
3$, while it is NP-complete for constant $\ell \ge 4$. We also show that the
problem is W[1]-hard parameterized by pathwidth${}+|A|$, while it is
fixed-parameter tractable parameterized by treewidth${}+\ell$.
