  Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J.
Keith and F. Zanello discovered infinite families of Ramanujan type congruences
modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19,
23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan
type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$.
In this paper, we introduce new infinite Ramanujan type congruences modulo $2$
for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao
and involve primes in $\mathcal P=\{p \text{ prime } : \exists \, j\in
\{1,4,8\},\, x, y \in \mathbb Z,\, \gcd(x,y)=1 \text { with } x^2+216y^2=jp\}$
whose Dirichlet density is $1/6$. As a key ingredient in our proof we show that
of the number of primitive solutions for $x^2+216y^2=pm$, $p \in \mathcal P$,
$p\nmid m$ and $pm\equiv 1\pmod{24}$, is divisible by $8$. Here, the difficulty
arises from the fact that $216$ is not idoneal. We also give a conjectural
exact formula for the number of solutions for this Diophantine equation. In the
second part of the article, we study reversals of Euler-type identities. These
are motivated by recent work of the second author on a reversal of Schur's
identity which involves $3$-regular partitions weighted by the parity of their
length.
