  We initiate the study of the cycle structure of uniformly random parking
functions. Using the combinatorics of parking completions, we compute the
asymptotic expected value of the number of cycles of any fixed length. We
obtain an upper bound on the total variation distance between the joint
distribution of cycle counts and independent Poisson random variables using a
multivariate version of Stein's method via exchangeable pairs. Under a mild
condition, the process of cycle counts converges in distribution to a process
of independent Poisson random variables.
