Great posts @6JRT's I can see how Parson would be good for fox hunting and Jack for badger. They are still very similar but have distinct type.
Some Parson breeders sell the runts of their bitches litters as Jacks, which means when they are bred with a JRT the pups will have both Parson & Jack genes in them, resulting in square back JRT's & not the oblong back JRT's