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In this correspondence, we determine the optimal power allocation, that achieves any specified point on the boundary of the capacity region, for sets of parallel Gaussian multiple-access channels (MACs) and sets of parallel Gaussian broadcast channels (BCs). The power allocation is across the parallel channels and, within each channel, across the users. In both cases, there is a single constraint on the total power used. First, the allocation for the parallel MACs is determined, in a simple form, using the Karush-Kuhn-Tucker (KKT) conditions and a simple Lagrangian argument. Using this result, the allocation for the parallel BCs is derived using recent findings on the duality of the MAC and BC.