In practical engineering applications, thermal protection structures are inevitably subjected to damage and failure. Meanwhile, the reliability analysis of these structures is challenged by various sources of uncertainty, including material dispersibility and load fluctuations. To address this challenge, this study proposes a hybrid random-interval reliability analysis method that simultaneously deal with both aleatory and epistemic uncertainties. Initially, the finite element method (FEM) is employed to simulate two typical failure modes: panel damage and interface delamination. Subsequently, random theory is applied to model the damage behavior of the panels, while interval analysis is used to quantify uncertainties in material properties and load boundaries. Building on these methodologies, a hybrid random-interval reliability analysis model is developed for the corrugated sandwich thermal protection structure. In addition, to improve computational efficiency, a Kriging surrogate model, optimized using a particle swarm optimization (PSO) algorithm, is utilized to replace time-consuming finite element simulations, enabling rapid evaluation of the structural limit state function. Finally, the proposed method is applied to assess the reliability of a honeycomb corrugated core thermal protection system. The results demonstrate that the new algorithm provides a fast and effective approach for evaluating the reliability of thermal protection structures under damage conditions.