Brian Rolston also scored in regulation for the Devils, who have won 14 of 17.
Tampa Bay got two first-period goals from Ryan Malone. The Lightning have lost eight of 10.
Parise tied it at 2 on a rebound goal during a power play at 1:13 of the third. The left wing has 35 goals this season.
Malone put the Lightning ahead 1-0 when he wrapped a shot from the slot around Devils defenseman Mike Mottau and past goalie Kevin Weekes at 4:33 of the first. He made it 2-0 on a wrist shot from the top of the left circle with 7:05 left in the period.
Rolston cut the Devils' deficit to 2-1 with a power-play goal at 2:16 of the second. His shot from just inside the blue line trickled past Tampa Bay goalie Karri Ramo into the net.
Weekes stopped Evgeny Artyukhin's shot on a breakaway midway through the third.
Tampa Bay is 7-22 in games decided by one goal this season. The Lightning are just 11-6-6 when ahead after the first period.
Notes: Injured New Jersey G Martin Brodeur (left elbow) made the trip to Florida and watched the game from the Devils' booth in the press box. Brodeur, who last played on Nov. 1, has been practicing with the team. ... Tampa Bay defensemen Paul Ranger and Andrej Meszaros are scheduled to undergo left shoulder surgery on Mar. 3. Recovery time is expected to be within six months. ... Devils RW Brendan Shanahan returned after missing two games with a bruised right foot. ... Lightning LW Gary Roberts (maintenance) didn't play.
