The Jazz fall to 0-3 for the first time since 1979, its first season in Utah.
"We've got to get better," said head coach Tyrone Corbin. "We're a young group and we should have more energy. We'll go back to work. It's frustrating have to learn lessons over and over again."
Jeremy Lin added 20 points as the Rockets (3-0) outscored the Jazz 64-37 in the second half.
After John Lucas made a jumper to beat the shot clock and pull the Jazz within 95-90 with 3:27 left, the Rockets allowed just one field goal the rest of the way and Harden ran the pick-and-roll repeatedly down the stretch.
"Each and every game we're getting better and better, but we've got to close out games," said Lucas. "That's the biggest thing, we've got to close games out. All these games, we've had right there in our hands, but we let them slip through our fingers."
Enes Kanter made an inside jumper to give Utah its final lead, 85-83 with 7:11 remaining. Harden drove for a layup, Lin hit a 3-pointer and Parsons followed with another basket to start the game-ending run.
Richard Jefferson led Utah with 18 points and Kanter added 16 for the Jazz, who looked disjointed once the Rockets increased their defensive pressure in the second half.
The loss marked the first time the Jazz have started a season with three consecutive defeats since they went 0-4 in the 1979-80 season, the year the franchise moved to Utah.
Harden made two free throws to cap a 15-2 run in the third quarter that brought the Rockets all the way back and knotted the game at 60-60. Dwight Howard had 10 and Harden had nine in the quarter in helping Houston outscore Utah 33-17.
Parsons, who also grabbed 12 rebounds, kept the Rockets in the game with 20 points in the first half, when the Jazz outrebounded Houston 27-12 and dominated the paint.
The Jazz led 56-37 just before the half, following a stretch that featured a dunk by Gordon Hayward and a layup by Derrick Favors that brought the crowd to its feet.
Howard had 15 points but was neutralized by Utah's young frontcourt of Kanter and Favors most of the game.
But Utah had no answer for the Rockets' defensive pressure and Harden's playmaking in the second half.
Once Harden started breaking down Utah's defense, the Rockets began finding open shots all over the court. Once the Jazz began missing shots, Houston was able to unleash its fast break and went 23 for 36 in final two quarters.