Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual $$L_2(\Omega)$$ inner product is replaced by an $$L_2(\Omega)\oplus L_2(\partial\Omega)$$ inner product. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semilinear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretization by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration.