Like you, I read the book carefully. If you're just arguing that Halberstam would suggest that "they can't work together" in the sense that they can no longer work under the same roof, i'd agree with you. But, otherwise, i think that Halberstam takes a much more nuanced view of the relationship than that and that it reflects the relationship itself.
It is, of course, naive to suggest that there weren't major issues between them when they went their separate ways, but the relationship endures in a transformed manner, much as many relationships between powerful people change after periods of rivalry and even intense anger. Parcells has frequently been quoted in recent years as saying that he and bb are in regular contact and that he has taken his advice since he began his Miami gig.
Here was Halberstam's take on the relationship as he presented it in an interview with Michael Holley after his book appeared. I think that the divorce analogy is apt; these are two people who can no longer live under the same roof but who can still have a relationship that is built on their common interests and past associations, as well as on mutual respect:
"they came together and it really worked. Parcells knew how to run a team, how to get the emotions, and Belichick was the creator of those great defenses with great players. They stayed together too long. It got harder and harder, I think, the more Belichick got to parity. The more people began to give him credit, I think, the more it grated on Parcells. And they probably should have gotten divorced about two years before they did."