Evan Rodrigues scored two goals, Mikko Rantanen had a goal and an assist, and Alexander Georgiev made 39 saves for Colorado (4-4-1), which was coming off a 1-0 loss to the New Jersey Devils on Friday.
"It's disappointing," Avalanche coach Jared Bednar said. "They turned up the dial in a bunch of different ways."
Parise cut it to 3-2 at 3:35 of the third period when he jammed in a rebound five-hole on Georgiev.
"I wish I had a couple (goals) back," Georgiev said. "(They) scored a couple that I wish I had played better on."
Mayfield tied it 3-3 at 10:39 when he redirected in a diagonal cross-ice pass from Mathew Barzal, and 17 seconds later, Beauvillier put the Islanders in front 4-3 when he shot short side on Georgiev from the left circle on a 2-on-1.
"Everyone kept pushing," Beauvillier said. "Everyone believed we could come back, and (it was) just one of those games where you get momentum after a goal and then keep pushing, pushing, pushing."