The image in the moduli space of curves of strata of meromorphic and quadratic differentials

We compute the dimension of the image of the forgetful map from $Z$ to the moduli space of curves, where $Z$ is a connected component of the stratum of $k$-differentials with an assigned partition $\mu$, for the cases when $k=1$ with meromorphic partition and $k=2$ when the quadratic differentials have at worst simple poles.