In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Peclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.