Both players ended regulation with scores of 19-under-par 261, after Leonard shot a final round 5-under 65, and Parnevik, the third round leader by three strokes, managed a 1-under 69.
He suffered his second playoff loss on the PGA Tour.
Leonard earned his 11th tournament title on the tour and first since 2005.
"That putt on 16 in the playoff, I thought I hit a good putt it just moved a little bit more than I thought," Leonard, a Dallas native, admitted. "Then I get up (to the 17th green) and I've got probably the easiest putt on this green. So it feels great to win."