  Graphical parking functions, or $G$-parking functions, are a generalization
of classical parking functions which depend on a connected multigraph $G$
having a distinguished root vertex. Gaydarov and Hopkins characterized the
relationship between $G$-parking functions and another vector-dependent
generalization of parking functions, the $\boldsymbol{u}$-parking functions.
The crucial component of their result was their classification of all graphs
$G$ whose $G$-parking functions are invariant under action by the symmetric
group $\mathfrak{S}_n$, where $n+1$ is the order of $G$. In this work, we
present a 2-dimensional analogue of Gaydarov and Hopkins' results by
characterizing the overlap between $G$-parking functions and 2-dimensional
$\boldsymbol{U}$-parking functions, i.e., pairs of integer sequences whose
order statistics are bounded by certain weights along lattice paths in the
plane. Our key result is a total classification of all $G$ whose set of
$G$-parking functions is $(\mathfrak{S}_p \times \mathfrak{S}_q)$-invariant,
where $p+q+1$ is the order of $G$.
