In this numerical study, we examined solid-state lasers that comprise side pumping with a blue light-emitting diode (LED) and a Ce:Nd:YAG lasing medium, utilizing the Monte Carlo photon tracing technique. We designed and modeled new cavities for LED side-pumped solid-state lasers. As a precondition for the experimental realization of these proposed cavities, we investigated trapezoidal and hexagonal shapes from the family of planar structures, along with parabolic and double parabolic shapes from the family of curved structures. Exploring a range of cavity parameter spaces, we determined the optimized parameters for the proposed cavities. Notably, the double parabolic cavity exhibited the highest efficiency in terms of laser performance. We calculated laser outputs by considering model parameters, such as pumping efficiency and absorption distributions, using space-dependent rate equations. Our study demonstrates the potential to achieve an output higher than 30 W with a 10% optical-to-optical efficiency for a blue-LED pumped Ce:Nd:YAG laser, and an even higher output, ∼40 W, with a slightly improved 13% efficiency for a near-infrared LED pumped Nd:YAG laser