Searching for a first win in four games, the Sea Eagles were 20-0 up by that point and Parramatta were heading for a sixth straight loss.
Stewart, playing his first game at Brookvale Oval since round seven, scored one try and was heavily involved in at least four more, including two in the last five minutes to Jamie Lyon and Kieran Foran.
He ran for almost 150 metres while wingers Taufua and David Williams both racked up more than 200 metres.
With their forwards dominant and playmakers Foran and Daly Cherry-Evans always probing, Manly had too many weapons for the tackle-shy Eels.
Manly could have won by an even bigger margin but bombed two tries with knock-ons and had two more rubbed out by the video referee.
Williams crossed in the third minute and Taufua got his first try in the ninth.
Taufua powered over for his second try off a Stewart pass and the fullback then crossed for one of his own.
Parramatta showed some fight in the final five minutes of the first half despite losing centre Jacob Loko to a patella injury.
Peni Terepo scored a try and Pat O'Hanlon was held up.
Manly lost lock Glenn Stewart after he was inadvertently poked in the eye by a teammate.
Even that setback had a silver lining for the Sea Eagles.
Stewart's replacement Tom Symonds scored a try within a minute of taking the field, though the latter appeared to suffer a thumb injury.
Parramatta winger Brayden Wiliame capitalised on a pass from halfback Chris Sandow for the Eels' second try.
It was one-way traffic after that with a try to Jamie Buhrer triggering a final-quarter flurry by the home team.
Parramatta coach Ricky Stuart said he was disappointed with the Allgood incident, but acknowledged punching wasn't allowed in the game.
"It escalated when Steve (Matai) didn't need to be there and Mitch didn't need to do what he did," Stuart said.
"I've already spoken to him and he thought he was going to get hit and he reacted.
"It was disappointing to see it happen, I don't want to condone that behaviour."
Manly coach Geoff Toovey was reluctant to comment on the incident.
"It was a bit of a fight, a bit of a dust-up, I'm sure they (the NRL) will sort it out," Toovey said.
He said Symonds suffered a broken thumb, prop George Rose twisted an ankle and Glenn Stewart got a poke in the eye.
He described Brett Stewart's involvement as excellent but said his side could improve.
Stuart said his players really tried despite the scoreline but made fundamental defensive errors.
"You can't have an error of judgment and you can't have a missed tackle and we had too many of both," Stuart said.
- AAP
