In the current article, we use a random supercontinuum based on a random Raman distributed feedback laser to investigate the generation of random numbers by spectrally demultiplexing the broad supercontinuum spectrum in parallel channels. By tuning the spectral separation between two independent channels, we test the most typically used statistical tests’ abilities to identify the required minimum spectral separation between channels, especially after the use of post-processing steps. Out of all the tests that were investigated, the cross-correlation across channels using the raw data appears to be the most robust. We also demonstrate that the use of post-processing steps, either least significant bits extraction or exclusive-OR operations, hinders the ability of these tests to detect the existing correlations. As such, performing these tests on post-processed data, often reported in literature, is insufficient to properly establish the independence of two parallel channels. We therefore present a methodology, which may be used to confirm the true randomness of parallel random number generation schemes. Finally, we demonstrate that, while tuning a single channel’s bandwidth can modify its potential randomness output, it also affects the number of available channels, such that the total random number generation bitrate is conserved.