It's the third straight conference finals appearance for San Antonio, which lost a heartbreaking seven-game series to the Miami Heat in last season's NBA Finals.
San Antonio, which had the NBA's best record, will await the winner of the series between Oklahoma City and the Los Angeles Clippers. The Thunder have a 3-2 lead.
LaMarcus Aldridge had 21 points for Portland, while Damian Lillard added 17 points and 10 assists.
The Trail Blazers' bench contributed only seven points compared to 40 by the Spurs' reserves.
After playing with various injuries throughout last postseason, the Spurs had been healthy in these playoffs until Wednesday.
Parker, who missed 13 games during the regular season, exited in the first half with tightness in his left hamstring.
After returning to the court with 9:46 remaining in the second quarter, Parker left the game exactly a minute later. He headed back to the locker room followed closely by San Antonio general manager R.C. Buford and team physician, Dr. Paul Saenz.
Parker was scoreless in 10 minutes, missing his only two shots as Portland pushed to keep the All-Star point guard out of the paint.
Manu Ginobili handled most of the ball-handling duties with Parker out.
Even with Parker slowed, the Spurs still had another good start thanks to their hustle on the boards.
Tiago Splitter had two offensive rebounds in the opening 3 minutes. The Spurs, who were averaging 9.3 offensive rebounds in the postseason, had five in the first quarter alone.
The Blazers went on a 9-0 run once Splitter exited, with Aldridge benefiting the most from the big man's absence. He hit a pair of jumpers over Boris Diaw for his first points of the game to spark the run.
Mills provided a huge spark even before Parker's exit.
The energetic Aussie sprinted for a layup after tipping away C.J. McCollum's dribble in the backcourt. A possession later, Mills blocked Lillard's 3-pointer, igniting a fast break that led to Green's first 3 of the game.
Green had the strongest effort of the postseason, going 4 for 6 on 3s.
San Antonio had four 3s in the second quarter, with Leonard's second extending its lead to 43-30 with 4:47 remaining in the half.
NOTES: Portland G Mo Williams (groin) missed his third straight game. After scoring six points on 3-for-11 shooting in the series opener, Williams exited Game 2 after nine minutes with a strained groin and has not played since. . Meb Keflezighi, the first American man to win the Boston Marathon in 31 years, was in attendance. . Ginobili slapped the scorer's table twice in celebration after Green's jumper rattled in with 6 minutes remaining. Green has struggled shooting throughout the postseason. .. Mills rushed down the court to hit a 3-pointer after receiving a push by Spurs assistant coach Sean Marks after the diminutive guard leapt into the team's bench to save a defensive rebound. . Spurs coach Gregg Popovich screamed at Duncan for not providing weak-side help for Diaw on a dunk by Aldridge with 6 minutes left in the first half. After initially shaking his head and frowning at Popovich, Duncan raised his hand, nodded his head and said ''My fault. My fault.'' Duncan later pointed to the sky after making a straightaway, 18-foot jumper with 5:24 remaining in the third quarter.