Inverse design has revolutionized the field of photonics, enabling automated development of complex structures and geometries with unique functionalities unmatched by classical design. However, the use of inverse design in nonlinear photonics has been limited. In this work, we demonstrate quantum and classical nonlinear light generation in silicon carbide nanophotonic inverse-designed Fabry-Pérot cavities. We achieve ultra-low reflector losses while targeting a pre-specified anomalous dispersion to reach optical parametric oscillation. By controlling dispersion through inverse design, we target a second-order phase-matching condition to realize second- and third-order nonlinear light generation in our devices, thereby extending stimulated parametric processes into the visible spectrum. This first realization of computational optimization for nonlinear light generation highlights the power of inverse design for nonlinear optics, in particular when combined with highly nonlinear materials such as silicon carbide.