Let me elaborate on my comment, adding some details and references.
The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.
Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.
Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbb{Q})$ admits a pure Hodge structure of weight $k$.