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.
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.
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 consecutive 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 further since 1981, when Danny Ainge was on the team.
Next up is a second-round game Saturday against No. 2 seed Kansas State.
KANSAS STATE 82, NORTH TEXAS 62
In Oklahoma City, Denis Clemente had 17 points and six assists, helping No. 2 seed Kansas State rout North Texas.
Kansas State (27-7) gathered itself after a shaky start, riding its defense into the second round for the second time in three years. The Wildcats held the Mean Green (24-9) to 31 percent shooting and stymied Josh White's forays into the lane, holding North Texas' leading scorer to three points.
Jacob Pullen and Curtis Kelly added 15 points each, and Kansas State had a 46-21 advantage inside to avoid joining the list of opening-day upset victims.
Next up is No. 7 seed BYU and sharpshooter Jimmer Fredette in the second round of the West Regional. The Wildcats should have an advantage, too; while they cruised in the first round, BYU needed two overtimes to beat Florida.
BUTLER 77, UTEP 59
In San Jose, Calif., Shelvin Mack's second-half surge helped Butler keep the nation's longest winning streak intact.
Mack hit a career-high seven 3-pointers and scored 18 of his 25 points after halftime, leading fifth-seeded Butler over UTEP.
Gordon Hayward added 13 points and Matt Howard had 11 as the Bulldogs (29-4) won their 21st game in a row to extend the sport's longest active winning streak -- but only after a dynamic rally from a six-point halftime deficit.
The Miners (26-7) were a popular NCAA upset pick after the Conference USA power received a surprisingly low seed in the tournament. Instead, the Bulldogs made a 22-4 run after halftime led by Mack, who hit five 3-pointers in the first 11 minutes.
Butler advanced to face 13th-seeded Murray State, which surprised Vanderbilt 66-65 on a buzzer-beater by Danero Thomas.
Derrick Caracter had 20 points and nine rebounds for UTEP, which won 16 consecutive games and a league title before a tournament loss to Houston last Saturday.