The set of parking functions of length $n$, $\operatorname{PF}_n$, is a generalization of the symmetric group on $n$ elements, first introduced by Konheim and Weiss in 1966. While much is known about these objects from an enumerative combinatorics perspective, far less is known from a probabilistic perspective. In this work, we give definitions of \textit{fast} and \textit{slow} single site dynamics on the set of parking functions, and compare the mixing times of these processes to the analogous process on the full space $[n]^n$. As far as we are aware, these are the first Markov chains to be defined that act directly on $\operatorname{PF}_n$, rather than sampling by way of a bijection. Moreover, as the set of parking functions is a monotone subset of $[n]^n$, determining the mixing time of these processes has interesting implications for the theory of Markov chain mixing, and more specifically, the cutoff phenomenon.