This paper studies mean square stabilization of MIMO discrete-time linear time-invariant systems over a parallel MIMO additive white noise (AWN) channel. Such channel consists of multiple independent SISO AWN channels subject to independent input power constraints. We derive dynamic output feedback stabilization conditions subject to such constraints, for minimum phase plants that have relative degree one. Consequently, these conditions are also valid for general plants provided that state feedback is employed. Our results show that necessary and sufficient condition for stabilization can be determined by studying the feasibility of a constrained algebraic Riccati equation. These results are also extended to stabilization problems subject to channel capacity constraints.