In this talk, I wish to report some work on applications of the $(p, k)$-parametrization of theta functions in the theory of integer partitions. We confirmed two conjectures on congruences for overpartitions and $k$-colored generalized Frobenius partitions by using the $(p, k)$-parametrization of theta functions. Those conjectures were given by Hirschhorn and Sellers, and Baruah and Sarmah, respectively. Moreover, we proved several conjectures of Sun on the relations between sums of squares and sums of triangular numbers.