While I don't believe that coaches can have an extreme negative effect on how well a running back performs, I do think that coaches can cause them to be more analytical of how they run with the ball. During Jones' first two injury-plagued seasons, he seemed to run less tentatively during games. Last season, he appeared to be more conscious of maintaining possession of the ball.
Parcells is a ball-control perfectionist. Fumbling is not optional. During 2006, I saw Jones covering up the ball very often with both arms right after the handoff and before he attempted to hit the hole. That's not what I remember of his running style in '04 and '05. He didn't first protect the ball and secondly find a running lane. He simply took the handoff and how he held the ball was an afterthought.
I think this had some impact on how he ran the ball last season and I put the 'blame' on Parcells. It's not a bad practice for a coach to instill, but it can hamper a running back's ability to process what his initial movement will be before the handoff or at the time of the handoff.
Will he second guess how he's going to plant his foot on a counter play? Is he determined to solely hit the hole if the play dictates it and disregard any option of taking a cutback lane if it catches the corner of his eye in that split second? Etc.
IMO, Jones is more a finesse back than a power back. He's more conscious of making a cut than securing the football. Instead of addressing that flaw in his game without reducing his overall effectiveness, I think that Parcells told Jones to hold onto the ball at all costs. That messed with his head. Take care of the ball first. Run second.
I've stated before on the forum that another similar running back, Tiki Barber, was well-known as a fumbler until Tom Coughlin's arrival. Coughlin (or someone on his staff that analyzed his problem) recognized that Barber would run with the ball away from his body. Coughlin corrected Barber's fumblitis by having him tuck the ball up high and into his chest. With his arm securing the rest of the ball underneath, it reduced fumbles caused by defenders (like Ellis) who knew that it was easy to strip the ball from Barber by simply grabbing at it as he carried it like a tote bag.
Afterwards, Barber's change in holding the ball did not impede his running style and his turnovers were reduced. The one positive thing that (I hope) will come about in this coaching change is that Jones will be helped in the same way. I want the return of the finesse back who ran the ball so well in limited games. Just my two cents.