The prediction of the mechanical response of polymers undergoing thermo-oxidation requires the simulation of the multi-physics chemical kinetic model that couples oxygen diffusion with non-linear reactions, referred to in the literature as the mechanistic model of thermo-oxidation (Colin et al. 2020). It involves species evolving over widely different timescales (up to 14 orders of magnitude difference in kinetic coefficients scales) and exhibiting strong coupling, resulting in high numerical stiffness. Consequently, discretization-based solvers recommended for such simulations (Colin et al. 2004) require high computational times and are thus limited to offline simulations for a single set of boundary conditions, prohibiting real-time simulation. Recent advances in physics-informed machine learning (PIML) (Toscano et al. 2025) offer a promising alternative for constructing surrogate models of physical systems without relying on labeled data. However, their application to stiff, coupled multi-physics problems remains challenging. In this work, we propose a physics-informed neural network (PINN) (Raissi et al. 2019) framework to obtain a real-time parametric surrogate of the thermo-oxidation mechanistic model for a family of boundary conditions without requiring training data. In order to mitigate the multi-scale difficulty and imitate the multi-physics coupling, the proposed model uses multiple neural networks coupled at the level of their physics-based residual loss. The boundary conditions are parametrized, enabling the PINN to act as a parametric surrogate rather than a single-solution solver. Additionally, a customized residual-based adaptive resampling strategy for domain collocation points is implemented, achieving improved accuracy compared with regular sampling methods. The model is evaluated based on its ability to reproduce derived macroscopic quantities and experimental observables, as well as species concentration profiles. Its computational performance is also assessed relative to the literature-recommended discretization-based solver. Finally, a transfer learning strategy is demonstrated, achieving up to four times reduction in training time when adapting the model to a new set of kinetic parameters.