Contextual Stochastic Bilevel Optimization (CSBO) extends standard Stochastic Bilevel Optimization (SBO) by incorporating context-specific lower-level problems, which arise in applications such as meta-learning and hyperparameter optimization. This structure imposes an infinite number of constraints - one for each context realization - making CSBO significantly more challenging than SBO as the unclear relationship between minimizers across different contexts suggests computing numerous lower-level solutions for each upper-level iteration. Existing approaches to CSBO face two major limitations: substantial complexity gaps compared to SBO and reliance on impractical conditional sampling oracles. We propose a novel reduction framework that decouples the dependence of the lower-level solution on the upper-level decision and context through parametrization, thereby transforming CSBO into an equivalent SBO problem and eliminating the need for conditional sampling. Under reasonable assumptions on the context distribution and the regularity of the lower-level, we show that an $ε$-stationary solution to CSBO can be achieved with a near-optimal sampling complexity $\tilde{O}(ε^{-3})$. Our approach enhances the practicality of solving CSBO problems by improving both computational efficiency and theoretical guarantees.