Brock Nelson scored twice and Scott Mayfield had a goal and an assist to help the Islanders get their third win in four games and eighth in the last 10. Sebastian Aho had two assists and Ilya Sorokin made 26 saves.
In the extra period, Parise rifled a high shot past Columbus netminder Joonas Korpisalo for his fifth goal of the season and the 413th of his career, sending the home crowd into a frenzy. Parise took a cross-ice pass from Jean-Gabriel Pageau after untangling himself from a Columbus player in the neutral zone. A penalty was about to be called as Parise scored.
''That goal is scored because of will and his work ethic,'' Islanders coach Lane Lambert said of the 38-year-old Parise. ''A great play by him.''
New York improved to 6-3-0 at home. A year ago, in their first season at UBS Arena, the Islanders started 0-5-2 on home ice.
''Everybody here has a belief that we can come back in any situation. I thought we did a good job of sticking with it,'' Nelson said. ''That's a good character win.''
Emil Bemstrom had a goal and an assist, and Marcus Bjork and Cole Sillinger also scored for the Blue Jackets, who have lost six of their last seven. Jake Bean and Gustav Nyquist each had two assists and Joonas Korpisalo finished with 41 saves.
''The guys played great in front of me. It was a tight game,? Korpisalo said. ''We were in it the whole time.''
Bjork, making his NHL debut, fired a shot past Sorokin at 9:38 of the third period to put Columbus ahead 3-2.
''The guys gave it their all tonight. I'm not going to get negative on that one at all,'' Blue Jackets coach Brad Larsen said. ''We played hard. Our team worked.''
Mayfield then tied it with 7:09 remaining with his fourth as the Islanders rallied from behind for the third time of the night.
''There's no give-up in this room,'' Mayfield said. ''It's nice to get the two points in the end.''
After a scoreless first period, Sillinger got the Blue Jackets on the scoreboard at 3:09 of the second with a power-play goal for his first of the season.
Nelson tied it at 7:27 before Bemstrom put the Blue Jackets back ahead a little more than three minutes later with his first.
Nelson got his second of the night and eighth of the season with 46 seconds left in the middle period to tie it 2-2.
The Islanders have outscored opponents 26-14 in the third period. They have combined for 27 goals in the first two periods and overtime.
The Blue Jackets are dealing with injuries to key players, most notably Zach Werenski, their top defenseman and power-play point man, who is out for the season because of a separated shoulder and torn labrum suffered in a win over Philadelphia on Thursday.
Columbus also placed D Nick Blankenburg (fractured ankle) and F Sean Kuraly and F Jakub Voracek (upper body injury) on injured reserve on Friday. Voracek is out indefinitely while Blankenburg will miss six-to-eight weeks. D Eric Gudbrandson also missed this game with an injury.
On Hockey Fights Cancer Night at UBS Arena, the Islanders remembered former players Clark Gillies, Mike Bossy and Jean Potvin before the game. All three died from cancer within the past year. Gillies and Bossy were linemates on the Islanders teams that won four straight Stanley Cups from 1980-83.
Islanders F Matt Martin played his 700th game with the franchise.
Islanders: At Ottawa on Monday night.
Blue Jackets: Host Philadelphia on Tuesday night.
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