Random number generation is a key element of stochastic simulations. It has been widely studied for sequential applications purposes, enabling us to reliably use pseudo-random numbers in this case. Unfortunately, we cannot be so enthusiastic when dealing with parallel stochastic simulations. Many applications still neglect random stream parallelization, leading to potentially biased results. Particular parallel execution platforms, such as Graphics Processing Units (GPUs), add their constraints to those of Pseudo-Random Number Generators (PRNGs) used in parallel. It results in a situation where potential biases can be combined to performance drops when parallelization of random streams has not been carried out rigorously. Here, we propose criteria guiding the design of good GPU-enabled PRNGs. We enhance our comments with a study of the techniques aiming to correctly parallelize random streams, in the context of GPU-enabled stochastic simulations.