Distributions of parity differences and biases in partitions into distinct parts

For a partition $\lambda ⊢n$, we let $\mathrm{pd}(\lambda )$, the parity difference of $\lambda$, be the number of odd parts of $\lambda$ minus the number of even parts of $\lambda$. We prove for $c_0\in \mathbb{R}$ an asymptotic expansion for the number of partitions of $n$ into distinct parts with normalised parity difference $n^{-1\mathrm{/}4}\mathrm{pd}(\lambda )$ greater than $c_0$ as $n\to \mathrm{\infty }$. As a corollary, we find the distribution of the parity differences and parity biases for partitions of $n$ into distinct parts. We also establish analogous results for generalised parity differences modulo $N$.