  A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density
$1/2$. Despite the remarkable amount of work of the last several decades,
however, even showing this density is positive seems out of reach. In a 2018
paper with Judge, we introduced a different approach and conjectured the
"striking" fact that, if for any $A \equiv \pm 1\ (\bmod 6)$ the multipartition
function $p_A(n)$ has positive odd density, then so does $p(n)$. Similarly, the
positive odd density of any $p_{A}(n)$ with $A\equiv 3\ (\bmod 6)$ would imply
that of $p_3(n)$.
  Our conjecture was shown to be a corollary of an earlier conjecture of the
same paper. In this brief note, we provide an unconditional proof of it. An
important tool will be Chen's recent breakthrough on a special case of our
earlier conjecture.
