Arithmetic properties of multiplier polynomials for certain polynomial maps

We investigate the arithmetic properties of the multiplier polynomials for certain $1$-parameter families of polynomials. In particular, we answer the unsolved Excercise 4.12 (c) of the book ``The Arithmetic of Dynamical Systems" by J. H. Silverman for broader classes of polynomials than quadratic polynomials. As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of unicritical polynomials. Moreover, candidates for the parabolic parameter are determined. We have also confirmed that it is a parabolic parameter, except $c=-3/2$.