  For a given prime $p$, we study the properties of the $p$-dissection
identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively.
Then as applications, we find many infinite family of congruences modulo 2 for
some $\ell$-regular partition functions, especially, for $\ell=2,4,5,8,13,16$.
Moreover, based on the classical congruences for $p(n)$ given by Ramanujan, we
obtain many more congruences for some $\ell$-regular partition functions.
