Model order reduction plays an important role in rendering multi-query scenarios for parametric problems computationally feasible. In this contribution we consider parametrized optimal control problems with a quadratic objective functional subject to linear time-varying dynamics. For the particular class of problems we are interested in, it is possible to derive a reduced model based on an approximation of the optimal final time adjoint. However, the online phase of this reduced model still involves computations in the high-dimensional state space of the control system. We therefore apply an additional reduction step based on reduced bases for the primal and adjoint systems. We derive a residual-based a posteriori error estimator for the fully reduced model and prove its reliability and further describe how to evaluate the fully reduced model as well as its error estimator efficiently. In addition, we propose different algorithms to construct the involved reduced bases and verify their performance on a benchmark example.