Concurrent estimation of muscle activations, joint contact forces, and joint kinematics by means of gradient-based optimization of musculoskeletal models is hindered by computationally expensive and non-smooth joint contact and muscle wrapping algorithms. We present a framework that simultaneously speeds up computation and removes sources of non-smoothness from muscle force optimizations using a combination of parallelization and surrogate modeling, with special emphasis on a novel method for modeling joint contact as a surrogate model of a static analysis. The approach allows one to efficiently introduce elastic joint contact models within static and dynamic optimizations of human motion. We demonstrate the approach by performing two optimizations, one static and one dynamic, using a pelvis-leg musculoskeletal model undergoing a gait cycle. We observed convergence on the order of seconds for a static optimization time frame and on the order of minutes for an entire dynamic optimization. The presented framework may facilitate model-based efforts to predict how planned surgical or rehabilitation interventions will affect post-treatment joint and muscle function.