I always thought the Parsons Russell (below) was pretty close to the original breed , long legged so able to keep up with the hounds/horses, yet small enough to go to ground.
Whereas the (English) Jack Russell is short legged and bred to kill its quarry rather than bay.
Either way I think the Parsons russell was a name invented by the american kennel club to allow KC registration for the breed in the early 2000s.
Clearly, I have no idea.