  We continue our study of a basic but seemingly intractable problem in integer
partition theory, namely the conjecture that $p(n)$ is odd exactly $50\%$ of
the time. Here, we greatly extend on our previous paper by providing a
doubly-indexed, infinite framework of conjectural identities modulo 2, and show
how to, in principle, prove each such identity. However, our conjecture remains
open in full generality.
  A striking consequence is that, under suitable existence conditions, if any
$t$-multipartition function is odd with positive density and $t\not \equiv 0$
(mod 3), then $p(n)$ is also odd with positive density. These are all facts
that appear virtually impossible to show unconditionally today.
  Our arguments employ a combination of algebraic and analytic methods,
including certain technical tools recently developed by Radu in his study of
the parity of the Fourier coefficients of modular forms.
