We consider the notion of classical parking functions by introducing
randomness and a new parking protocol, as inspired by the work presented in the
paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by
Carlson, Christensen, Harris, Jones, and Rodr\'iguez. Among our results, we
prove that the probability of obtaining a parking function, from a length n
preference vector, is independent of the probabilistic parameter p. We also
explore the properties of a preference vector given that it is a parking
function and discuss the effect of the probabilistic parameter p. Of special
interest is when p=1/2, where we demonstrate a sharp transition in some
parking statistics. We also present several interesting combinatorial
consequences of the parking protocol. In particular, we provide a combinatorial
interpretation for the array described in OEIS A220884 as the expected number
of preference sequences with a particular property related to occupied parking
spots, which solves an open problem of Novelli and Thibon posed in 2020
(arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena
in combinatorics and provide further directions for research.Comment: 22 pages, 3 figures, 4 table