"We expect it," Byrd said. "He's unbelievable how he comes through. It seems like every single day we jump on his back."
The Mets, who have won two in a row after dropping six straight, scored four of their first five runs on homers -- including a solo shot by Byrd in the eighth -- before taking a two-run lead off Jordan Walden (1-1) in the 10th.
But it was Wright's seventh homer that stunned the crowd at Turner Field and helped hand the Braves a third straight loss and an 11th defeat in their last 16 games.
Even Wright seemed surprised that Kimbrel, who had converted 98 of 110 career save opportunities, left a fastball up in the strike zone.
"He's as dominant as anybody in the game," Wright said. "You've just got to hope he makes a minor mistake or really you don't have much of a chance."
Added Mets manager Terry Collins, "I'm not sure how many hits we've ever gotten off Kimbrel, let alone a home run."
Collins then pulled Parnell and brought in pinch-hitter Mike Baxter in the middle of the at-bat.
After Walden hit Baxter with a pitch, Ruben Tejada followed with an RBI single to score Valdespin from second and make it 6-5.
Baxter, who advanced to third on Tejada's hit, scored on Daniel Murphy's RBI single off Luis Avilan for a 7-5 lead. Jeurys Familia earned his first save by retiring facing the minimum in the bottom half of the 10th.
Walden gave up two runs while getting just two outs.
Parnell (2-0) got two outs for the win.
Braves starting pitcher Mike Minor retired the last 18 batters he faced, but left for a pinch-hitter in the seventh with the score tied 3-all.
Minor didn't receive a decision after Byrd's solo homer off Eric O'Flaherty in the eighth made it 4-4.
Byrd helped boost New York's confidence, too, especially against an Atlanta bullpen that led the majors with a 1.94 ERA and was tied for the best opponents' batting average at .208.
"When O'Flaherty came in," Collins said, "you were hoping you were going to scratch something out."
Minor allowed three hits and three runs with no walks and four strikeouts. He left for a pinch-hitter as Tyler Pastornicky bunted in the bottom of the seventh when the Braves took their first lead at 4-3.
Minor, who gave up a season-high six runs in his last start, an 8-3 loss last Sunday at Detroit, and the left-hander got off to rough start against New York.
The Mets took a 2-0 lead in the first on John Buck's 10th homer.
Lucas Duda made it 3-0 in the second with his sixth homer, an opposite-field shot to left-center, but then Minor settled down.
Francisco left the game one out later when he suffered a mild right ankle sprain while leading off second and was caught stealing with Schafer attempting to bunt. Pena replaced Francisco at third base.
Marcum gave up six hits, three runs, three walks and struck out four in 4 1/3 innings.
In the seventh, Scott Atchison walked Pena, who moved to second on a sacrifice bunt and advanced to third on a wild pitch.
Schafer drew his fourth walk, a career high, before Simmons drove in Pena to make it 4-3 with a fielder's choice RBI.
But it was Wright, a six-time All-Star and the fourth team captain in franchise history, who delivered the big blow that stirred the Mets' dugout.
"There's a reason why No. 5's a star," Collins said. "He comes up with big things at big times."
Braves manager Fredi Gonzalez said C Brian McCann, who's on the disabled list following offseason surgery to repair a torn labrum, could be activated Monday or Tuesday at Cincinnati. ... The Mets optioned CF Collin Cowgill and called up Andrew Brown from Triple-A Las Vegas before the game. Brown went 0 for 3 and started in right. Byrd started in center. ... Since coming off the disabled list last Aug. 24 with Milwaukee, Marcum is 2/3 with a 5.05 ERA in 10 starts and one relief appearance.