This paper presents an auto-tuning framework for torque-based Nonlinear Model Predictive Control (nMPC), where the MPC serves as a real-time controller for optimal joint torque commands. The MPC parameters, including cost function weights and low-level controller gains, are optimized using high-dimensional Bayesian Optimization (BO) techniques, specifically Sparse Axis-Aligned Subspace (SAASBO) with a digital twin (DT) to achieve precise end-effector trajectory real-time tracking on an UR10e robot arm. The simulation model allows efficient exploration of the high-dimensional parameter space, and it ensures safe transfer to hardware. Our simulation results demonstrate significant improvements in tracking performance (+41.9%) and reduction in solve times (-2.5%) compared to manually-tuned parameters. Moreover, experimental validation on the real robot follows the trend (with a +25.8% improvement), emphasizing the importance of digital twin-enabled automated parameter optimization for robotic operations.