Parametric PDE models with high-dimensional parameter spaces can be computationally costly to solve, particularly in the presence of strongly nonlinear boundary conditions. This bottleneck is especially severe when a parameter within the boundary condition approaches a singular limit, for example, as in Robin to Dirichlet limits. In this regime, full-order solves remain costly, while modern global operator surrogates such as neural operators lose robustness near the singular limit and lack reliability guarantees. We deal with this problem through selective boundary condition reduction, a hybrid model reduction method that reduces only the boundary/interface law to an asymptotic, simpler boundary condition, and only when warranted. For each parameter vector input, a neural network predictor estimates the interior and trace errors induced by replacing the full boundary condition with its singular limit boundary condition. The substitution by the asymptotic reduced model is accepted only if both predicted errors are below a tolerance defined by the user; otherwise, the computation reverts to the full model. Training uses paired offline solves (full law vs. limit law) to define supervised interior/trace error targets, and we incorporate the a priori knowledge of singular-limit convergence rates in order to accelerate training. We evaluate our approach on parametric PDEs with Robin and highly nonlinear Neumann boundary conditions that exhibit singular-limit behavior, motivated by electrochemistry applications. The method delivers significant speedups over classical state-of-the-art methods while maintaining high reliability relative to modern surrogate approaches. Overall, selective boundary condition reduction emerges as a robust and practical model reduction strategy that combines singular limit analysis with machine learning to improve efficiency without sacrificing reliability.