Let \(p(n)\) denote the number of partitions of \(n\), let \(1 < D \equiv 23 \pmod{24}\) be square-free and let \(q := e^{2\pi i z}\). As an application of generalized Borcherds products arising from weight \(1/2\) harmonic Maass forms, the author shows that \[ \widehat{F}(D;z):= \sum_{m \geq 1 \atop \gcd(m,6) = 1} p\left( \frac{Dm^2+1}{24}\right) \sum_{n \geq 1 \atop \gcd(n,D) = 1} q^{mn} \] is congruent modulo \(2\) to a certain weight \(2\) meromorphic modular form. Employing Galois representations and the local nilpotency of the Hecke algebra modulo \(2\), the author then shows that if there exists an \(m\) coprime to \(6\) such that \[ p\left( \frac{Dm^2+1}{24}\right) \] is even (resp. odd), then there exists infinitely many such \(m\). He gives upper bounds in terms of the class number \(h(-D)\) for the first such \(m\) (if one exists) and leaves open the conjecture there are always such \(m\). He establishes the conjecture in the even case for certain \(D\), such as \(D\) prime.