Proofs of some conjectures of Keith and Zanello on $t$-regular partition

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$.