Blake, one of two Islanders who collected a career-high four points, started and finished the scoring for New York. The Islanders chased star goaltender Nikolai Khabibulin after two periods.
Khabibulin surrendered six goals on 32 shots after yielding just six goals in his previous three games. He has played in 18 of Tampa Bay's last 19 games.
Parrish scored in each of the first two periods, including a one-on-one move past defenseman Dan Boyle before stickhandling around Khabibulin that made it 4-1.
It was one of four second-period goals by the Islanders, who are unbeaten in their last three games (2
After Parrish netted his second goal at the 13-minute mark, Dave Scatchard and Jason Wiemer tallied in a 19-second span to stretch the lead to 6-1.
Wiemer also picked up three assists against his former team for his first career four-point effort.
Tampa Bay's Ben Clymer opened the scoring and teammate Andre Roy capped it, but the Lightning suffered their fifth straight loss against the Islanders.