In 1967, Aronson and Serrin proved the parabolic Harnack inequality for weak solutions of a class of quasilinear parabolic equations on Euclidean space. This talk is about generalizing their result to metric spaces. In the special case of the linear heat equation, a result of Grigor'yan, Saloff-Coste (and Sturm) states that the parabolic Harnack inequality is equivalent to the volume doubling property and the Poincar\'e inequality, as well as to two-sided Gaussian heat kernel estimates. I will introduce a notion of a 'quasilinear form' on metric measure Dirichlet spaces which resembles the structural conditions on the quasilinear equation proposed by Aronson and Serrin. Assuming volume doubling and Poincar\'e inequality, we will obtain the parabolic Harnack inequality for quasilinear forms. If time permits, I might also discuss applications to estimating Dirichlet heat kernels.