The Gators, the two-time defending NCAA champions, watched an early 17-3 lead evaporate, falling behind on Jerren Shipp's 3-pointer with 13:30 to go. But Florida (24-11) responded with a 15-6 run and put the game away on Parsons' 3-pointer with 2:32 to play.
Nick Calathes had 11 points and nine assists for Florida. The Gators shot a season-high 60.5 percent from the floor and picked apart Arizona State's zone defense with 26 baskets on 23 assists.
James Harden scored 18 points for ASU (21-13).
The Gators will play Massachusetts in the semifinals in Madison Square Garden on April 1.
Last March, the Gators were steaming toward their second straight NCAA title. On Tuesday night, they found themselves in the desert against the motivated Sun Devils, who were still angry over their snub by the NCAA tournament selectors.
The game matched longtime friends and colleagues Billy Donovan of Florida and Herb Sendek of Arizona State. Sendek served on Rick Pitino's Providence staff when Donovan played for the Friars, and then the two served on Pitino's staff at Kentucky.
Tensions ran high despite the friendship. At one point, Donovan was speaking to an official during a timeout when an incensed Sendek dropped his clipboard and stormed toward center court, where he was intercepted by another official.
Both teams stumbled down the stretch in the regular season but turned it up in the NIT. Florida won its first two tourney games by an average of 26 points, and ASU by 11 and 14 points.
Early on, the Gators looked like the hotter team, sprinting to a 14-point lead after six minutes. The Gators hit four 3-pointers during that flurry, two by Hodge. The Gators shot 61.9 percent from the floor in the first half, and 46.2 percent from beyond the arc.
Arizona State went on an 8-2 run midway through the half and whittled the lead to 35-29 at halftime.