In this talk I will discuss how to extend the natural action of S_n on parking functions of length n, to an action of S_n+1. This includes, as a special case, the fact that the Lie representation, studied by Stanley, Sundaram and others, extends the regular representation of S_n to a representation of S_n+1. The representations involved arise as Macaulay inverse systems to some naturally defined ideals, I also hope to discuss a few things related to this. This is joint work with Brendon Rhoades (UCSD).