  We continue a program of investigating the parity of the coefficients of
eta-quotients, with the goal of elucidating the parity of the partition
function. In this paper, we consider positive integer powers $t$ of the
Dedekind eta-function. Previous work and conjectures suggest that arithmetic
progressions in which the Fourier coefficients of these functions are even
should be numerous. We explicitly identify infinite classes of such
progressions modulo prime squares for several values of $t$, and we offer two
broad conjectures concerning their existence in general.
