Dirk Nowitzki's 21-foot jumper with 16.6 seconds left pushed the Dallas lead to 112-109, and the Mavericks held on to snap a three-game losing streak.
Deron Williams scored 22 points and Nowitzki finished with 18.
Marc Gasol led Memphis with 22 points, while Zach Randolph had 20 points and 14 rebounds.
The teams were tied at 102 at the end of regulation. The Mavericks had the last shot with less than a second left, but failed to convert a lob at the rim to Justin Anderson.
In overtime, Parsons' 3-pointers countered any Memphis baskets inside by Gasol. Parsons' second long-range conversion gave Dallas a 110-106 lead with 2:24 left in overtime.