We present a novel algorithm to determine the payoff-space of certain normal-form C1 parametric games, and - more generally - of continuous families of normal-form C1 games. The algorithm has been implemented by using MATLAB, and it has been applied to several examples. The implementation of the algorithm gives the parametric expressions of the critical zone of any game in the family under consideration both in the bistrategy space and in the payoff space and the graphical representations of the disjoint union (with respect to the parameter set of the parametric game) of the family of all payoff spaces. We have so the parametric representation of the union of all the critical zones. One of the main motivations of our paper is that, in the applications, many normal-form games appear naturally in a parametric fashion; moreover, some efficient models of coopetition are parametric games of the considered type. Specifically, we have realized an algorithm that provides the parametric and graphical representation of the payoff space and of the critical zone of a parametric game in normal-form, supported by a finite family of compact intervals of the real line. Our final goal is to provide a valuable tool to study simply (but completely) normal-form C1-parametric games in two dimensions.