Parker’s transport equation stochastic solution for simulation cosmic rays distribution in the heliosphere is demanding on computing resources. Simulations can last days, weeks, or even months with certain input parameters. We implemented 1D Forward-in-time and Backward-in-time models for GPU with successful acceleration ranged from 7x to 86x. This acceleration was gained with not a negligible reduction of accuracy, especially with changing the entire simulation from double-precision float-point format to floating-point format. This led to a certain deviation that we called pulsations that showed in results with input time step less than 2.0 s. In this paper, we discuss the parallelization process on GPU. We also discuss the comparison of our solution with Dunzlaff et al. and the overall accuracy of results gained from GPU implementation of 1D Forward-in-time and Backward-in-time models.