Bunning, then of the Philadelphia Phillies, spun his June 21, 1964, perfecto against the New York Mets at Shea Stadium -- on Father's Day.
For both men, the timing of the rare feats of athletic perfection was particularly poetic and emotional.
Braden lost his single mother, Jodie Atwood, to cancer while a senior in high school in rough-and-tumble Stockton, Calif., but his grandmother, Peggy Lindsey, has been there every step of the way for the A's southpaw, and she made the hour drive to the Coliseum on Sunday.
As soon as Braden got Gabe Kapler to ground out to shortstop to complete the 27-up, 27-down, 109-pitch, 77-strike, six-strikeout gem, Lindsey was waiting for Braden on the field and the two shared a long, poignant embrace.
"It's a more important day for my grandmother than anything," Braden said. "That's the biggest thing to be able to give her something like this on a day of this magnitude, considering everything we've been through together. It's more about her for me.
"[Mother's Day] hasn't been a joyous day for me in a while. But to know that I still get to come out and compete and play a game on that day, that makes it a little better. With my grandma in the stands, that makes it a lot better. To be able to give her this today was perfect."
Father's Day in 1964 was perfect for Bunning, the right-hander who would go 19-8 with a 2.63 ERA that year. In front of 32,026 fans at Shea, Bunning, a father of seven, negotiated the 27 outs in only 90 pitches, throwing 79 of them for strikes and fanning 10 batters as the Phillies won, 6-0, in the first game of a doubleheader.
It was the first National League perfecto in 84 years, and it was accomplished even though by all accounts, Bunning didn't adhere to much superstition.
"He was really silly," Bunning's catcher that day, Gus Triandos, told the Bowling Green (Ky.) Daily News in 1989 on the 25th anniversary of the game. "He was jabbering like a magpie."
Bunning reportedly had no problems mentioning that he had a no-hitter going while sitting in the dugout, a strict baseball no-no. But he also helped himself to the victory, doubling in two runs in the game.
And in the ninth inning, he still had a lighthearted approach to the proceedings, calling Triandos to the mound to have the backstop tell him a joke.
"I couldn't think of anything," Triandos told the Bowling Green Daily News. "I just laughed at him."
In a special Father's Day touch to rival that of Braden's tearful Mother's Day hug with his grandma, it just so happened that Bunning's gem was twirled in front of his wife, Mary, and 12-year-old daughter, Barbara, who had driven up to Queens from their Cherry Hill, N.J., home and were seated in the stands right behind home plate.
"I knew what I had done," Bunning told the Bowling Green Daily News. "But not necessarily the historical significance."
Bunning ended up on "The Ed Sullivan Show" that night, would go on to win 100 games in each league and be elected into the Hall of Fame. He's currently a United States Senator from Kentucky.
After all he now has in common with Bunning, maybe Braden has Letterman, Cooperstown and politics in his future.