In this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations. We consider first the case of box constraints on the control in a general setting and then, in the case of a quadratic Hamiltonian, we also impose final integral constraints on the state. For the first problem, in a rather general framework, we prove a characterization of the quadratic growth property in the strong sense, i.e., for admissible controls whose associated states are uniformly near to the state of the nominal control. Assuming a quadratic Hamiltonian in the state constrained case, we provide a sufficient second order optimality condition for the aforementioned quadratic growth property, which does not impose the uniqueness of the Lagrange multiplier set. As a consequence of our results, in the quadratic Hamiltonian case, under some continuity assumptions on the data we prove that the notions of quadratic growth in the strong sense coincide with the more standard notion of quadratic growth in the weak sense, i.e., with respect to controls which are uniformly near to the nominal one.