  Parking functions of length $n$ are well known to be in correspondence with
both labelled trees on $n+1$ vertices and factorizations of the full cycle
$\sigma_n=(0\,1\,\cdots\,n)$ into $n$ transpositions. In fact, these
correspondences can be refined: Kreweras equated the area enumerator of parking
functions with the inversion enumerator of labelled trees, while an elegant
bijection of Stanley maps the area of parking functions to a natural statistic
on factorizations of $\sigma_n$. We extend these relationships in two principal
ways. First, we introduce a bivariate refinement of the inversion enumerator of
trees and show that it matches a similarly refined enumerator for
factorizations. Secondly, we characterize all full cycles $\sigma$ such that
Stanley's function remains a bijection when the canonical cycle $\sigma_n$ is
replaced by $\sigma$. We also exhibit a connection between our refined
inversion enumerator and Haglund's bounce statistic on parking functions.
