The parity conjecture for even and odd 3-regular partitions on \mathcal P-progressions

Let $b_{3,e}(n)$ and $b_{3,o}(n)$ denote the numbers of $3$-regular partitions of $n$ having, respectively, an even and an odd number of parts. Let $n\geqslant 0$, let $p\in\mathcal P$ satisfy $p\equiv 1,7\pmod{24}$, and let $0\leqslant\alpha<p$ with $\alpha\ne\lfloor p/24\rfloor$. Here $24_p^{-1}$ denotes the inverse of $24$ modulo $p^2$. The parity conjecture. One has

$$b_{3,e}\bigl(2(p^2n+p\alpha-24_p^{-1})\bigr)\equiv 0\pmod 2,$$ $$b_{3,o}\bigl(2(p^2n+p\alpha-24_p^{-1})\bigr)\equiv 0\pmod 2.$$

The conjecture is proposed as an analogue of the paper's main congruence theorem, with numerical evidence mentioned but no proof given.