We present an open-source software package, TRAVOLTA (Terrific Refinements to Accelerate, Validate, and Optimize Large Time-dependent Algorithms), for carrying out massively parallelized quantum optimal control calculations on GPUs. The TRAVOLTA software package is a significant overhaul of our previous NIC-CAGE algorithm and also includes algorithmic improvements to the gradient ascent procedure to enable faster convergence. We examine three different variants of GPU parallelization to assess their performance in constructing optimal control fields in a variety of quantum systems. In addition, we provide several examples with extensive benchmarks on our GPU-enhanced TRAVOLTA code to show that it generates the same results as previous CPU-b...