The BYU star told Parsons, the only Gators player to ever hit two true buzzer-beaters to clinch victories, "no game winning shots tonight."
Fredette went on to score 37 points and hit two 3-pointers in double overtime to seal the Cougars' 99-92 win over 10th-seeded Florida on Thursday, the first time they've reached the second round of the tournament in 17 years.
Parsons, meanwhile, missed potential game-winners at the end of regulation and the first overtime, and Fredette didn't let Florida get another chance for a dramatic finish.
"We had a second life, or a third life, or whatever it was, and I just wanted to go out there and try to get it done because I don't know if I could have played another overtime," said Fredette, the nation's 12th-leading scorer.
Fredette made a 3-pointer from the top of the key to put seventh-seeded BYU (30-5) ahead 90-83 with 2:42 left, then added another from the right wing with just over a minute remaining to put the game away.
Fredette surpassed the 30-point plateau for the eighth time this season and the third straight game, coming off 45- and 30-point performances in the Mountain West tournament.
Michael Loyd Jr. added a career-high 26 points, pushing BYU to a lead it would hold until a late second-half rally by Florida. Loyd also scored all six of the Cougars' points in the first overtime and hit a 3-pointer to start the second overtime that put BYU ahead to stay.
"He was ready and his number was called and he came through for us," coach Dave Rose said of Loyd, who had been averaging 4.6 points and reached double figures just five times.
Kenny Boynton scored 27 points for the Gators (21-13), helping to rally them from a 13-point second-half deficit in their first NCAA tournament game since back-to-back national titles in 2006 and 2007.
BYU snapped a streak of seven straight first-round losses in the tournament while ending Florida's 12-game NCAA winning streak. It's the Cougars' first win since 1993, and they haven't gone farther since 1981, when Danny Ainge was on the team.
"This was a long time in coming for our program, and it's a big win," said Rose, who was successfully treated for pancreatic cancer last June. "One of the most important goals we had at the start of this season was to get into this tournament and advance, and we're advancing. I'm proud of our guys."
Boynton rallied the Gators in the second half with a series of four 3-pointers, the last one putting Florida ahead 71-68 with 2:36 left in regulation -- their first lead of the second half.
After Vernon Macklin missed one of two free throws in the final minute, Fredette was able to tie it at 75 on a layup with 25.3 seconds left.
Parsons missed a jumper from the right side of the lane in the closing seconds, and Jonathan Tavernari missed the backboard entirely with a heave from beyond halfcourt after the Cougars called timeout with half a second left in regulation.
Parsons also missed a 3 in the last minute of the first overtime and Dan Werner missing an awkward volleyball-style tap off a busted play after Fredette had dribbled into Boynton's hands.
"You can't complain about the looks we got at the end," said Parsons, who hit game-winners at the final buzzer against North Carolina State and South Carolina earlier this season. "We fought hard, everyone was tired but they were just as tired as us."
Parsons finished with 20 points and 10 rebounds, Macklin scored 16 and Erving Walker added 12 points. Florida's hopes fizzled after Boynton fouled out after Loyd's 3-pointer to open the second overtime.
"That was a big play in the game when he fouled out and they had to play down the stretch without him because he was hitting 3s, he was penetrating to the basket, making free throws, and he was a heck of a defender," Rose said.
Rose was able to steal some quick rest for Fredette down the stretch, substituting for him when the Cougars went on defense. The few extra gasps helped, and Fredette felt a rush of adrenaline entering the second overtime after BYU had dodged elimination again and again.
BYU, the nation's third-best 3-point shooting team, was 3-for-3 from behind the arc in double overtime, with Fredette stopping after his second in a row to gesture to a pack of Cougars fans gathered behind the BYU bench. He clenched his left fist as he dribbled the ball upcourt to run out the final seconds.
"You're playing for your life," he said. "It's one and done at this point."