We present a parallel GPU-accelerated solver for branch Model Predictive Control problems. Based on iterative LQR methods, our solver exploits the tree-sparse structure and implements temporal parallelism using the parallel scan algorithm. Consequently, the proposed solver enables parallelism across both the prediction horizon and the scenarios. In addition, we utilize an augmented Lagrangian method to handle general inequality constraints. We compare our solver with state-of-the-art numerical solvers in two automated driving applications. The numerical results demonstrate that, compared to CPU-based solvers, our solver achieves competitive performance for problems with short horizons and small-scale trees, while outperforming other solvers on large-scale problems.