And it allowed them to escape playoff elimination with a 105-103 win over the Thunder on Monday night.
The Rockets led in the fourth quarter of each of the previous two games only to end up losing.
"We felt the pain and frustration from the last two losses ... and we didn't want that to happen for the third straight time," Parsons said.
It almost did.
A stunned Ibaka fell to the court after the miss and covered his face with his hands.
Durant scored 38 points in Oklahoma City's second game without injured All-Star guard Russell Westbrook.
The victory kept Houston from being swept in the playoffs for the first time since 1996. Game 5 is Wednesday in Oklahoma City.
"Getting this first win gives us confidence going back to Oklahoma City, and anything can happen," Harden said.
Rockets coach Kevin McHale was proud of his young team's work in getting its first playoff victory.
"That was a gutsy win," McHale said. "We're not going to lie down. The one constant has been their willingness to scrap and fight. We kept on fighting."
Durant hit the go-ahead 3-pointer in the Thunder's Game 3 win. He wanted to get another one off Monday, but Francisco Garcia forced him to give up the ball.
"I was. I was going to try to win it," Durant said. "But he played good defense."
Durant is looking forward to the series returning to Oklahoma City.
"We'll be all right," he said. "We're up 3-1 with a good opportunity to close it out at home. We couldn't ask for a better situation, so hopefully we'll get the job done."
Harden scored 15 points but also had 10 turnovers. He had two chances to extend Houston's lead with less than a minute left but missed both of them, including shooting an air ball.
Getting the win allowed Harden to take his tough night in stride.
"I did have a double-double," he deadpanned, referring to his points and turnovers.
He picked up his fifth foul with about seven minutes remaining, sending him to the bench. A dunk by Jackson seconds later cut Houston's lead to 98-94.
Jackson got the Thunder within 100-98 with a 3-pointer a couple of minutes later. His 3-point attempt on the next trip down the floor rattled in and out of the basket.
Houston scored four quick points to extend the lead to 104-98 before Harden returned to the game with about three minutes left.
He said he struggled to get going after sitting out in the fourth quarter.
"The foul trouble kind of set me back," he said. "Coming in trying to make big shots, I was cold."
Derek Fisher made a 3-pointer to cut Houston's lead to 104-101 with less than three minutes remaining. It was reviewed a few seconds later, and the points were taken away because replays showed the shot clock had expired.
Jackson finished with 18 points in his second start in place of Westbrook, and Kevin Martin added 16.
Oklahoma City's DeAndre Liggins received a technical foul with :07 left in the third quarter for arguing the call on a jump ball. Delfino made the free throw, but Durant hit a jump shot at the buzzer to get the Thunder within 91-84 entering the fourth quarter.
The Rockets led by two points in the third quarter before a 3-point play by Harden. He took Jackson off the dribble for a jump shot and drew the foul -- Jackson's fourth -- to send him to the bench and the crowd into a frenzy.
Durant was shaken up a few seconds later when he crashed into the scorer's table chasing a loose ball. He remained on his back for a couple of minutes holding his right wrist before being helped to his feet. But he didn't leave the game and seemed to be OK.
Harden's three-point play was the start of a 12-2 Houston run capped by another three-point play -- this one by Delfino -- to extend the lead to 85-75 with about three minutes left in the third quarter.
Delfino dunked after stealing the ball from Martin, and Durant fouled him on the play.
Houston scored the first 10 points of the second half to erase a halftime deficit and take a 63-60 lead. Parsons had six points in that span.
The Rockets did a much better job of keeping the crowd involved in this one than they did in Game 3, when they trailed early by as many as 26 points. Houston rallied to take a lead in the fourth quarter of that game before losing 104-101.
Near the end of the first half, Durant sailed into the crowd Superman-style to save the ball on a bad pass from Harden. He was able to get to his feet and stood in the crowd watching as Martin swished a 3 on the other end. He couldn't figure out how to get out of the crowd for a few seconds before finally climbing out and back onto the court. But before he got back to the game, Durant asked a woman in the first row whom he had crashed into whether she was OK.
Houston led by five at the end of the first quarter and was up by three points early in the second before Oklahoma City scored 12 straight points, fueled by seven from Durant. The Rockets took a 43-34 lead about 6 1/2 minutes before halftime.
Delfino swiped the ball from Durant and swished a 3 with 17 seconds left in the first half to cut the Thunder's lead to 60-53 at halftime.
The Rockets led 29-24 after one quarter thanks to 5-of-9 shooting from 3-point range Monday after trailing by 20 points after one in Game 3.
Houston's Greg Smith returned Monday after missing Game 3 with a stomach ailment. ... Durant has scored 20 or more points in 30 straight playoff games. ... Former Rockets star Steve Francis watched the game from a courtside seat. ... The Seattle SuperSonics were the last team to sweep Houston in the playoffs, getting a 4-0 series win in the Western Conference semifinals in 1996.