  A partition of $n$ is called a $t$-core partition if none of its hook number
is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a
parity result for $3$-core partition function $a_3(n)$. Motivated by this
result, both the authors \cite{MeherJindal2022} recently proved that for a
non-negative integer $\alpha,$ $a_{3^{\alpha} m}(n)$ is almost always divisible
by arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by
arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t=
p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ In this article,
by using Hecke eigenform theory, we obtain infinite families of congruences and
multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which
generalizes some results of Das \cite{Das2016}.
