Parker netted 20 points and handed out nine assists while Leonard connected on four 3-pointers en route to 18 points as San Antonio drained club playoff-record 14 treys in all.
Matt Bonner popped in 12 points off the bench for the Spurs, who led virtually the entire game and jumped off to a quick advantage in the 2013 Western Conference finals.
The Spurs were making their eighth appearance in the series in 15 years.
Leading Memphis scorer Zach Randolph was held to just two points on 1-of-8 shooting as the Grizzlies fell in their first-ever conference finals appearance.
Celebrity Couples of 2014 [PHOTOS]