When one studies the dynamics of a 1-parameter family of polynomial maps, parabolic parameters (i.e., the parameter for which the polynomial has a periodic point such that the multiplier is a root of unity) have important information about the family. Recently, we gave some integrality results of the multiplier polynomials for certain $1$-parameter families of polynomials. As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of uni-critical polynomials. Moreover, we determined the quadratic parabolic parameters for $z^2 + c$. We introduce these results in this talk. This talk is based on a joint work with Yuya Murakami and Kohei Takehira.