Neural Ordinary Differential Equations (NODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, requiring prohibitive time steps for explicit solvers. In this work, we introduce a novel data-driven reparametrization strategy to address the stiffness issue, enabling the efficient use of explicit solvers for stiff systems. Our approach constructs a data-driven time map based on the adaptive time-stepping of an implicit solver, transforming the original stiff system into a non-stiff one that is computationally efficient to solve. After solving the reparametrized system, we recover the original dynamics through a learned mapping via a neural network. We leverage this strategy to build ROMs for stiff systems while significantly reducing the computational cost. The proposed method is validated on several benchmark stiff ODEs, demonstrating substantial improvements in speed without compromising precision. The neural network model also showcases strong generalization properties beyond training times while maintaining robustness, accuracy, and consistency.