One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair (A, B) whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters that cause the resulting matrix pair to be controllable. In this paper, the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair (A, B). Subject to a certain condition on the parameterization called the "binary assumption," an explicit graph-theoretic characterization of such matrix pairs is derived.