Parker scored nine of his points in the first quarter as the Spurs jumped off to a 24-14 lead and never were headed en route to the win, which completed a 4-0 series sweep.
The Spurs point guard also dished off six assists while Tim Duncan contributed 15 points, eight rebounds and four blocks for San Antonio, which shot 51.3 percent from field in taking its first conference championship since 2007.
The second-seeded Spurs now will face the winner of the Miami-Indiana Eastern Conference series in the NBA Finals, where they will go for the franchise's fifth all-time league title.
The Heat hold a 2-1 advantage over the Pacers with Game 4 set for Tuesday at Indianapolis.
Quincy Pondexter netted 22 points for fifth-seeded Memphis, which fell in its initial conference finals appearance.