This work presents a systematic study of the preservation of supermodularity under parametric optimization, that allows us to derive complementarity among parameters and monotone structural properties of optimal policies in many operations models. We introduce new concepts of mostly-lattice and additive mostly-lattice, which significantly generalize the commonly imposed lattice condition, and use them to establish the necessary and sufficient conditions on the feasible set so that supermodularity can be preserved under various assumptions on the objective functions. We further identify some classes of polyhedral sets which satisfy these concepts. Finally, we illustrate how our results can be used on a two-stage optimization problem.