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 selfsimilarity
results for
${b}_{t}(n)$
modulo $2$ for
certain values of
$t$.
Further, they proposed some conjectures on selfsimilarities of
${b}_{t}(n)$
modulo $2$ for
certain values of
$t$.
For example, for a positive proportion of primes
$p$, they
conjectured that
${b}_{3}(n)$
satisfies
$$\sum _{n=0}^{\infty}{b}_{3}(2(pn+\alpha )){q}^{n}\equiv \sum _{n=0}^{\infty}{b}_{3}(2n){q}^{pn}\phantom{\rule{0.3em}{0ex}}(mod\phantom{\rule{0.3em}{0ex}}2),$$ 
where
$\alpha \equiv 2{4}^{1}\phantom{\rule{0.3em}{0ex}}(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}{p}^{2})$,
$0<\alpha <{p}^{2}$. In this paper, we prove
their conjectures on
${b}_{3}(n)$
and
${b}_{25}(n)$. We also prove a
selfsimilarity result for
${b}_{21}(n)$
modulo $2$.
The proofs use a result of Serre which says that if
$f$
is an integer weight cusp form on the congruence subgroup
${\Gamma}_{0}(N)$, then for any positive
integer
$M$, a positive
proportion of the primes
$p\equiv 1\phantom{\rule{0.3em}{0ex}}(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}M\u201cN)$
has the property that
$f(z)\mid {T}_{p}\equiv 0\phantom{\rule{0.3em}{0ex}}(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}M)$.
