As a matter of convention, we usually run 3 or 4 chains in JAGS. By default, this gives rise to chains that draw samples from 3 or 4 distinct pseudorandom number generators. I didn’t go and check whether it does things 111,222,333 or 123,123,123, but in any event the “parallel chains” in JAGS are samples

Zero Inflated Models and Generalized Linear Mixed Models with R.
Zuur, Saveliev, Ieno (2012).