Distributions of parity differences and biases in partitions into distinct parts

For a partition $\lambda ⊢n$, we let $pd\left(\lambda \right)$, the parity difference of $\lambda$, be the number of odd parts of $\lambda$ minus the number of even parts of $\lambda$. We prove for $c_0\in R$ an asymptotic expansion for the number of partitions of $n$ into distinct parts with normalised parity difference $n^{-1/4}pd\left(\lambda \right)$ greater than $c_0$ as $n\to \infty$. 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 $N$.