  The action of the symmetric group $S_n$ on the set $Park_n$ of parking
functions of size $n$ has received a great deal of attention in algebraic
combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an
action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module
$V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $Park_n$. We
describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and
describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We
give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose
representation theory is governed by a bivariate generalization of Dyck paths.
A Fuss generalization of our results is a special case of this bivariate
generalization.
