Parity results for certain partition functions and identities similar to theta function identities

In this paper we give a collection of parity results for partition functions of the form \[ \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2 \] and \[ \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2 \] for various sets of positive integers S, which are specified with respect to a modulus, and quadratic polynomials $e(n)$ and $f(n)$. Several identities similar to theta function identities, such as \[ \prod _{\substack {n = 1\\n \nequiv \pm (4,6,8,10)\pmod {32}}}^\infty (1 - x^n) = 1 + \sum _{n = 1}^\infty (-1)^n (x^{n^2} + x^{2 n^2}), \] and its associated congruence \[ \prod _{\substack {n = 1\\n \nequiv 0, \pm 2, \pm 12, \pm 14,16 \pmod {32}}}^\infty (1 - x^n)^{-1} \equiv 1 + \sum _{n = 1}^\infty (x^{n^2} + x^{2 n^2}) \pmod 2, \] are also proved.