This paper is concerned with an optimal control problem governed by a nonsmooth, semilinear parabolic PDE. The nonlinearity in the state equation is only directionally differentiable, locally Lipschitz continuous and is allowed to have infinitely many non-differentiable points. By employing its limited properties, Bouligand-differentiability of the control-to-state map is shown (in an extended sense). This enables us to establish second-order sufficient optimality conditions. We provide concrete settings where these reduce to the first-order necessary optimality condition.