Let the class BPNC (the combination of $\mathsf{BPP}$ and $\mathsf{NC}$) be log depth parallel algorithms with bounded error probability and access to a random source (I'm not sure if this has a different name). Define the class DBPNC similarly, except that all processes have random access into a random stream of bits fixed at algorithm startup.
In other words, each process in BPNC has access to a distinct random source, while DBPNC algorithms have a shared perfectly random counter mode generator.
Do we know whether BPNC = DBPNC?