We propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case. The method, which is based on an off-line–on-line decomposition procedure, allows at the on-line step large computational cost savings with respect to the “truth” approximation used for defining the reduced basis. An a posteriori error estimate is provided by means of the goal-oriented analysis, thus associating an error bound to each optimal solution of the parametrized optimal control problem and answering to the demand for a reliable method. An adaptive procedure, led by the a posteriori error estimate, is considered for the generation of the reduced basis space, which is set according to the optimal primal and dual solutions of the optimal control problem at hand.