This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations. The state and co-state are approximated by Raviart- Thomas mixed finite element spaces, and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. A priori error estimates are derived for the state, the co-state, and the control. Some numerical examples are presented to confirm the theoretical investigations.