Parsons also distributed eight assists as Houston stayed alive after coming into Game 5 of their Western Conference quarterfinal series trailing the Thunder 3-0.
The Rockets forward picked up the slack for leading scorer James Harden, who labored through an off-night with just 15 points on 4-of-12 shooting.
Houston hung on for the victory when Oklahoma City's Serge Ibaka missed a potential game-tying lay-in at the buzzer.
Kevin Durant poured in 38 points for Oklahoma City, which will again seek to close out the series at home in Wednesday's Game 5.
2014: The Year in Music [PHOTOS]