The Geometric Dynamical Northcott Property in the quadratic family

The aim of this note is to give a proof of Theorem A from our work on the geometric Northcott property in the simpler case of the quadratic family; being in dimension $1$ in both the dynamical space and the parameter space, and having a simple and explicit parametrization of the family allow to simplify the proof and, we hope, make the ideas more apparent.