I will present recent results on applying the parabolic Moser iteration method in the setting of (fractal-type) metric measure Dirichlet spaces. Under appropriate geometric conditions, we obtain that non-negative local weak solutions to the heat equation are locally bounded, Hölder continuous, and satisfy a strong parabolic Harnack inequality. The parabolic Harnack inequality is known to characterize heat kernel estimates of (sub-)Gaussian type. An application of this equivalence, together with the Doob's transform technique, allow to obtain sharp bounds for the Dirichlet heat kernel on inner uniform domains.