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

$$\prod\limits_{n \in S} {{{(1 - {x^n})}^{ - 1}} \equiv \sum\limits_{ - \infty }^\infty {{x^{e(n)}}\;\pmod 2} }$$

and

$$\prod\limits_{n \in S} {{{(1 - {x^n})}^{ - 1}} \equiv \sum\limits_{ - \infty }^\infty {({x^{e(n)}} + {x^{f(n)}})\;\pmod 2} }$$

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

$$\prod\limits_{\mathop {n = 1}\limits_{n \nequiv \pm (4,6,8,10)\;\... ...\mathop \sum \limits_{n = 1}^\infty {{( - 1)}^n}({x^{{n^2}}} + {x^{2{n^2}}}),}$$

and its associated congruence

$$\prod\limits_{\mathop {n = 1}\limits_{n \nequiv 0, \pm 2, \pm 12,... ...quiv 1 + \sum\limits_{n = 1}^\infty {({x^{{n^2}}} + {x^{2{n^2}}})\;} \pmod 2,}$$

are also proved.