"Whoever's the best team, I want to play them," Leonard said. "It's gonna prepare us for the (NBA) finals."
LaMarcus Aldridge had 21 points for Portland, while Damian Lillard added 17 points and 10 assists.
With Parker forced to exit in the first half with a hamstring injury, Leonard, Mills and Green pressed the action on both ends.
San Antonio finished with 13 steals, including five for Leonard, while forcing 18 turnovers.
"My mindset was just trying to be aggressive on the offensive end," Leonard said. "Just knowing Tony was out, he's very aggressive and still try to stick with our system. Play our offense, move the ball, just try to be a little more aggressive."
The Spurs also had 24 assists on 42 baskets, shooting 47 percent from the field, and had 43 fast-break points.
"They definitely showed us about moving the ball around," Aldridge said "They made five or six passes every possession down. It just makes your defense tired and just makes guys make mistakes. They definitely showed use where we are trying to go."
After playing with various injuries throughout last postseason, the Spurs had been healthy in these playoffs until Wednesday.
After returning to the court with 9:46 remaining in the second quarter following his normal rest, 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.
"Luckily we have a couple of days," Duncan said. "It worries us, obviously, but he's been going hard. He's had a great series thus far, and it just kind of caught up with him. They pulled the plug on him before he hurt himself. He started to feel a little weird and they did the right thing. Hopefully that helps; hopefully there's nothing there."
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.
With Parker out, the Spurs' bench outscored the Trail Blazers' 40-7.
"Everybody stepped up," forward Boris Diaw said. "We have a very solid team, a very deep bench."
The lack of bench production was especially critical considering how Splitter and the Spurs were able to contain Aldridge.
After averaging 29.8 points in Portland's upset of Houston in the opening round of the playoffs, Aldridge was limited to 21.8 points.
"I don't know maybe you can tell me," Aldridge said when asked about the Spurs' defense. "I had the same looks. I actually got to the rim more in this series than in last series. I just missed easy shots. I have to be better for us to win. I definitely didn't play well in this series so it was tough for us to win games."
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.
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.
"What Patty (Mills) did today was outstanding," Ginobili said. "Patty, Danny, and Kawhi, especially, were fantastic. We struggled all series in the third quarters, and they really stepped up. They were aggressive, getting steals, running the transition and ones we got that lead back to 20, it kind of felt like it was over."
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.