I do agree that what color's the perception of New England pre-Parcells is that whenever they had winning seasons, there was always some sort of "but..." associated with it.
1976: Ben Dreith derails what was NE's most talented team ever to that point in time. That team is probably up there in the Top 10, possibly Top 5, of all Pats teams, IMHO.
1978: Fairbanks fiasco.
Then begins the typical slide into mediocrity, which results in a disastrous 2-14 season just one year after missing the playoffs by 1 game.
The ride back up is a bit bumpy, with the Ron Myer/Rod Rust fiasco one year, but the result is the rebuilt Patriots making the SB by finally squishing the fish under Raymond Berry. Not only the Pats have the misfortune of facing one of the NFL's all time best teams in the Super Bowl, they themselves are one of the statistically worst teams to have made the SB by that point. The blowout loss is followed by back-biting and the drug scandal, and the predictable slide into mediocrity and worse begins.
The Patriots teams from 1989 through 1992 were terrible; the 1990 team was one of the worst in league history. The first round draft picks during that period brought the Patriots Hart Lee Dykes, Chris Singleton, Ray Agnew, Pat Harlow, Leonard Russell, and my all time favorite, Eugene Chung. Fans, who were always lukewarm towards attending games at the dump known as Shaeffer Stadium, were staying away. The most memorable televised home game was in 1990 against the Giants, when New York fans stacked the stadium to see their team struggle against the 1-14 Patriots. Add in the ownership situation, and the openly stated desire of the league to abandon New England altogether for a domed stadium in warmer climes, the future of the franchise looked bleak.
I agree that Parcells had little to do with fixing most of those problems. But he did draft well, and the team did improve, and there was at least an air of respectability about the team. Some of that was Orthwein, who, while obviously looking to sell/move the team, at least attempted to run the place like a NFL franchise instead of the clown circus it had become under the Sullivans and later Kiam. But, yes, Parcells did not save the day. But I do recall feeling a lot better after the Patriots flushed out the amateurs and hired Parcells; and then feeling even better when he drafted Bledsoe instead of Chung'ing the pick the way the local mediots wanted him to.
I do think the team's success under Kraft/Belichick/Brady, and the contributions of Parcells draftees such as Vinatieri, Ty Law, Tedy Bruschi and Willie McGinest in those Pats Super Bowl wins have contributed to fans forgetting some of the dark side of Parcells' tenure here. As for Carroll, he was in the no win situation to begin with, and the team slid to mediocrity thanks to some very bad drafts, and Carroll became the whipping boy for the team's overall lack of discipline (I still remember Chris Canty forgetting to pick up a loose ball because he was dancing after making a routine tackle in a blowout loss to Tampa). Some of that is on Carroll, but some of that was out of his control. No matter; all of the above stated problems have been long solved.