  A parking function of length n is a sequence (b_1, b_2,..., b_n) of
nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has
the property that a_i < i for every i. A well-known result about parking
functions is that the polynomial P_n(q), which enumerates the complements of
parking functions by the sum of their terms, is the generating function for the
number of connected graphs by the number of excess edges when evaluated at
(1+q). In this paper we extend this result to arbitrary connected graphs G. In
general the polynomial that encodes information about subgraphs of G is the
Tutte polynomial, which is the generating function for two parameters, namely
the internal and external activities, associated with the spanning trees of G.
We define G-multiparking functions, which generalize the G-parking functions
that Postnikov and Shapiro introduced in the study of certain quotients of the
polynomial ring. We construct a family of algorithmic bijections between the
spanning forests of a graph G and the G-multiparking functions. In particular,
the bijection induced by the breadth-first search leads to a new
characterization of external activity, and hence a representation of Tutte
polynomial by the reversed sum of G-multiparking functions.
