Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial ideal in the polynomial ring $S = {mathbb K}[x_1, dots, x_n]$ where a set of generators are indexed by the nonempty subsets of $[n] = {1,2,dots,n}$. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined by $M^{(k)}_n$, a subideal of $M_n$ obtained by allowing only generators corresponding to subsets of $[n]$ of size at most $k$. For each $k$ the set of standard monomials of $M^{(k)}_n$, denoted $text{stan}_n^k$, contains the usual parking functions and has interesting combinatorial properties in its own right. For general $k$ we show that elements of $text{stan}_n^k$ can be recovered as certain vector-parking functions, which in turn leads to a formula for their count via results of Yan. The symmetric group $S_n$ naturally acts on the set $text{stan}_n^k$ and we also obtain a formula for the number of orbits under this action. For the case of $k = n-2$ we study combinatorial interpretations of $text{stan}_n^{n-2}$ and relate them to properties of uprooted trees in terms of root degree and surface
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