  Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$
assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where
$0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By
proving Subbarao's Conjecture, Radu showed that there are no such congruences
when it comes to parity. There are infinitely many odd (resp. even) partition
numbers in every arithmetic progression. For primes $\ell \geq 5,$ we give a
new proof of the conclusion that there are infinitely many $m$ for which
$p(\ell m+\delta_{\ell})$ is odd. This proof uses a generalization, due to the
second author and Ramsey, of a result of Mazur in his classic paper on the
Eisenstein ideal. We also refine a classical criterion of Sturm for modular
form congruences, which allows us to show that the smallest such $m$ satisfies
$m<(\ell^2-1)/24,$ representing a significant improvement to the previous
bound.
