Summary: We define an extension of the standard binding number of a graph which introduces parameters into the computation. We call the result a parameterized binding number. This extension is motivated by a number of theorems that use bounds on the order of neighbor sets of vertices to determine the existence of cycles or factors within the graph. We demonstrate how this extended binding number can be integrated into such theorems. Additionally, we present theorems that provide sufficient conditions on the degree sequence of a graph which guarantees a prescribed lower bound on parameterized binding numbers. These degree sequence theorems are shown to be best possible in a certain sense. Finally, we show how these degree conditions can be combined with known theorems to produce sufficient conditions which guarantee certain cycles or factors within the graph.