Explicit stabilized integrators are an efficient alternative to implicit or semiimplicit methods to avoid the severe time-step restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.