  We find an explicit $S_n$-equivariant bijection between the integral points
in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the
permutahedron, and the set of $m$-parking functions of length $n$. This
bijection restricts to a bijection between the regular $S_n$-orbits and
$(m,n)$-Dyck paths, the number of which is given by the Fuss-Catalan number
$A_{n}(m,1)$. Our motivation came from studying tilting bundles on
noncommutative Hilbert schemes. As a side result we use these tilting bundles
to construct a semi-orthogonal decomposition of the derived category of
noncommutative Hilbert schemes.
