The difference is that the Rams are counting on Fisher being around for a long time and the fact that he was with his last team for 17 years shows he didn't take the Rams job only to leave in a few years. When Parcells joined the Dolphins no one thought he would be around real long and most didn't think he would stay his entire four year contract. He didn't.
The Rams are Fisher's team and the people hired around him will be there as long as he is. Parcells position was to build a front office and coaching staff that he felt comfortable with then turn it over to them. Obviously it did not go as smoothly as was planned but the plan was always to have Ireland take over full control of making roster decisions. Whoever the Rams hire will never be able to work their way up to that.