The Mersenne Twister (MT) is a pseudo-random number generator (PRNG) widely used in High Performance Computing for parallel stochastic simulations. We aim to assess the quality of common parallelization techniques used to generate large streams of MT pseudo-random numbers. We compare three techniques: sequence splitting, random spacing and MT indexed sequence. The TestU01 Big Crush battery is used to evaluate the quality of 4096 streams for each technique on three different hardware configurations. Surprisingly, all techniques exhibited almost 30% of defects with no technique showing better quality than the others. While all 106 Big Crush tests showed failures, the failure rate was limited to a small number of tests (maximum of 6 tests failed per stream, resulting in over 94% success rate). Thanks to 33 CPU years, high-quality streams identified are given. They can be used for sensitive parallel simulations such as nuclear medicine and precise high-energy physics applications.