Proof of the Ballantine-Merca Conjecture and theta function identities modulo 2

For positive integers $m$ we consider the theta functions $f_m(z):=\sum_{mk+1\text{ square }}q^k$. Due to classical identities of Jacobi, it is known that $$f_4\equiv f_6f_{12}\pmod 2.$$ Here we prove that the only triples $(a,b,c)$ for which $f_a\equiv f_bf_c\pmod 2$ are of the form $(2q,4q,4q)$ or $(4q,4q,8q)$, where $q$ is any positive odd number, or belong to the following finite list $$\{(4,6,12),(6,8,24),(8,12,24),(10,12,60),(15,24,40),(16,24,48),(20,24,120),(21,24,168)\}.$$ The result is inspired by the Ballantine-Merca Conjecture on recurrence relations for the parity of the partition function $p(n)$, which we also prove here.