  In a recent paper, Andrews and Newman extended the mex-function to integer
partitions and proved many partition identities connected with these functions.
In this paper, we present parity considerations of one of the families of
functions they studied, namely $p_{t,t}(n)$. Among our results, we provide
complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$.
