The
G
G
-parking function ideal
M
G
M_G
of a directed multigraph
G
G
is a monomial ideal which encodes some of the combinatorial information of
G
G
. It is an initial ideal of the toppling ideal
I
G
I_G
, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for
M
G
M_G
was given by Postnikov and Shapiro in the case when
G
G
is saturated, i.e., whenever there is at least one edge
(
u
,
v
)
(u,v)
for every ordered pair of distinct vertices
u
u
and
v
v
. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of
M
G
M_G
for any undirected multigraph
G
G
, as well as for a family of related ideals including the toppling ideal
I
G
I_G
. This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.