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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 ≢ 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. [ABSTRACT FROM AUTHOR]