We present a number of results on inverse parametric optimization and its application to hybrid system control. We show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem. We exploit these results in application to the control of systems with piecewise affine dynamics, and show that it is possible to model such systems as optimizing processes. Optimal control problems for such systems can be remodeled as bilevel optimization problems and solved with existing techniques.