Precise and efficient calculations are necessary to accurately assess the effects of thermal protection system (TPS) uncertainties on aerospacecrafts. This paper presents a probabilistic design methodology for TPSs based on physics-informed neural networks (PINNs) with parametric uncertainty. A typical thermal coating system is used to investigate the impact of uncertainty on the thermal properties of insulation materials and to evaluate the resulting temperature distribution. A sensitivity analysis is conducted to identify the influence of the parameters on the thermal response. The results show that PINNs can produce quick and accurate predictions of the temperature of insulation materials. The accuracy of the PINN model is comparable to that of a response surface surrogate model. Still, the computational time required by the PINN model is only a fraction of the latter. Considering both computational efficiency and accuracy, the PINN model can be used as a high-precision surrogate model to guide the TPS design effectively.