Let $p_{odd}(n)$ be the number of partitions of $n$ into odd parts (see here). For instance, one has the generating function $$\prod_{k\geq1}\frac1{1-q^{2k-1}}.$$

QUESTION.What is the size of this set $$A_N:=\{n\in\{1,2,\ldots,N\}: \text{$p_{odd}(n)$ is odd}\}$$ for large $N$?

**Note.** I do not expect this to be $\sim\frac12N$. Any solution or reference is appreciated.