That’s not what happened. BB was never hired by the Jets as a head coach, BB’s 1997 assistant coach contract said he’d become HC of the NYJ when Parcells left.
BB got tired of Parcells and wanted to go to NE. Since that would be a promotion the Jets would be unable to block it. So Parcells “retired” making BB the head coach by operation of contract (i.e. not by an actual hiring process), enabling the Jets to c*ckblock NE since it would now be a lateral move.
BB now really didn’t want to stay at that point because Parcells would be the GM and shadow coach and BB wouldn’t have the independence he wanted (in addition to the ownership change, but I think that was somewhat of a pretext and Parcells hanging around was the real reason). So he quit as HC of the NYJ, sued the Jets and NFL to be set free, lost, and was then eventually set free by Parcells after he squeezed a bunch of draft choices out of NE.