  We construct a family of $S_n$ modules indexed by $c\in\{1,\dots,n\}$ with
the property that upon restriction to $S_{n-1}$ they recover the classical
parking function representation of Haiman. The construction of these modules
relies on an $S_n$-action on a set that is closely related to the set of
parking functions. We compute the characters of these modules and use the
resulting description to classify them up to isomorphism. In particular, we
show that the number of isomorphism classes is equal to the number of divisors
$d$ of $n$ satisfying $ d\neq 2 \: (\!\!\!\!\mod 4)$. In the cases $c=n$ and
$c=1$, we compute the number of orbits. Based on empirical evidence, we
conjecture that when $c=1$, our representation is $h$-positive and is in fact
the (ungraded) extension of the parking function representation constructed by
Berget and Rhoades.
