Say what???...
They were mentor and protogee. Parcells always had an eye for talent on and off the field. He also had an ego that would not allow for crediting that talent and a penchaant for keeping it under his thumb - something Belichick has chosen not to do once in a position to because it was done to him. He and Bill had a functional relationship that spanned the Giants and Jets with a HC position for Belichick in Cleveland in between. He tried to keep Belichick in NY as a HC even as he feigned moving to the FO, but Belichick was tired of living in his shadow and knew that was where he would remain unless he cut the cord publicly and quite messily.
That was the end of the relationship until a couple of years ago when Parcell's was beginning to have his struggles reinventing himself as the genius in Dallas and his brother, who was his only real friend over the years, died of cancer. It was written about fairly extensively at the time... Parcells's began facing his own mortality and questioning a lifelong self-absorbed mindset that had cost him a marriage (wife divorced him after his first retirement on the heels of leaving the Jets once he lost Belichick) and alienated his children (Dallas and Scott went with Belichick when he left Parcells and Scott and Belichick became best friends - their kids didn't see much of grampy as a family as a result) as well as most of his former protogees and left him basically alone save for those who were using him as he used them (like Jerry Jones and his ESPN cronies). That is when he reached out to several former protogees including Belichick and asked if they could revisit the whole relationship thing. I think the only one he hasn't made up with is Charlie...whom he embarassed publicly while with the Giants and who testified against him when Belichick wanted out of that Jets contract.
One of the things Belichick learned from Parcells's was how not to alienate protogees needlessly while pushing them mercilessly. Even Mangini could have positioned himself to have an enduring relationship with Belichick once he moved on had his ego (much like Parcell's) allowed him to see that Belichick just wanted what was best for him, knew he wasn't ready to be a HC and knew the Jets job was a HCing disaster in waiting. Mangini isn't on Bill's **** list for leaving so much as leaving prematurely before paying his dues and taking a job he was advised to avoid for his own good and probably landing it because he was all too willing to be disloyal to his mentor and tell tales to a division rival that the coaching fraternity mandates remain off limits. Unlike Parcell's Belichick had done a lot to deserve loyalty including being personally financially generous to and supportive of his young protogees and finding ways to credit their contributions and advance their careers (where possible and deserved) even as he maintained the one voice philosophy within his own organization.
Parcell's seems to have learned that lesson finally as in Miami he is far more inclined to give credit to those he brought in to run the team on the field. He'll still pull some crap like he did over his contract terms this winter, because he still needs to see himself as a big blip on the NFL radar, but he is no longer inclined to grind up his coaching staff to achieve that end.