Estimating rate coefficients from complex chemical reactions is essential for advancing detailed chemistry. However, the stiffness inherent in real-world atmospheric chemistry systems poses severe challenges, leading to training instability and poor convergence, which hinder effective rate coefficient estimation using learning-based approaches. To address this, we propose a Stiff Physics-Informed Neural ODE framework (SPIN-ODE) for chemical reaction modelling. Our method introduces a three-stage optimisation process: first, a black-box neural ODE is trained to fit concentration trajectories; second, a Chemical Reaction Neural Network (CRNN) is pre-trained to learn the mapping between concentrations and their time derivatives; and third, the rate coefficients are fine-tuned by integrating with the pre-trained CRNN. Extensive experiments on both synthetic and newly proposed real-world datasets validate the effectiveness and robustness of our approach. As the first work addressing stiff neural ODE for chemical rate coefficient discovery, our study opens promising directions for integrating neural networks with detailed chemistry.