We present an optical approach for high-speed parallel random bit generation based on stochastic pulse-to-pulse fluctuation in the supercontinuum (SC). Through spectrally demultiplexing the SC pulse sequence into different wavelength channels, we simultaneously extract multiple independent fast random bit streams from each SC pulse subsequence via associated comparators in parallel. Proof-of-concept experiments demonstrate that using our method, four 10 Gb/s random bit streams are obtained from a SC pulse source with verified randomness. Moreover, this method also provides a promising strategy to fabricate ultrafast random bit generators with Tb/s throughput capacity just by increasing additional wavelength channels.