We first introduce SmoothedTurboMPC, a GPU-parallel differentiable MPC solver combining sequential quadratic programming (SQP) with a log-barrier-smoothed alternating direction method of multipliers (ADMM) inner solver. Its design enables smooth gradients, exact sensitivities at convergence, and efficient execution on GPUs. Second, to efficiently train MPC policies, we pair our solver with first-order reinforcement learning (FORL) using Short-Horizon Actor-Critic (SHAC), which reuses rollout MPC solutions instead of repeatedly re-evaluating them. Comparisons with state of the art solvers show that the approach achieves up to 4.4× higher end-to-end training throughput. On a vehicle racing task, our method achieves higher returns than MPC PPO with a 46× speedup, training on 3.3 million transitions in only 7 minutes. Finally, we use the framework to train a more reliable MPC racing policy via RL with domain randomization, and validate it on a full-scale race car. We integrate our advances into the open-source TurboMPC solver.