We study the generic dimension of the solution set over C^*, R^* and R_{>0} of parametric polynomial systems that consist of linear combinations of monomials scaled by free parameters. We establish a relation between this dimension, Zariski denseness of the set of parameters for which the system has solutions, and the existence of nondegenerate solutions, which enables fast dimension computations. Systems of this form are used to describe the steady states of reaction networks modeled with mass-action kinetics, and as a corollary of our results, we prove that weakly reversible networks have finitely many steady states for generic reaction rate constants and total concentrations.