Consider that a linear time-invariant (LTI) plant is given and that we wish to design a stabilizing controller for it. Admissible controllers are LTI and must belong to a pre-selected subspace that may impose structural restrictions, such as sparsity constraints. The subspace is assumed to be quadratically invariant (QI) with respect to the plant, which, from prior results, guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided. This paper addresses the previously unsolved problem of extending Youla's classical parametrization so that it admits QI subspace constraints on the controller. In contrast with prior parametrizations, the one proposed here does not require initialization and it does not require the existence of a stable stabilizing controller. The main idea is to cast the stabilizability constraint as an exact model-matching problem with stability restrictions, which can be tackled using existing methods. Furthermore, we show that, when it exists, the solution of the exact model-matching problem can be used to compute an admissible stabilizing controller. Applications of the proposed parametrization on the design of norm-optimal controllers via convex methods are also explored. An illustrative example is provided, and a special case is discussed for which the exact model matching problem has a unique and easily computable solution.