Parsons committed to the Grizzlies immediately after a Friday meeting at agent Dan Fegan's home in Beverly Hills, California.
The 27-year-old Parsons also met with and received a max contract offer from the Portland Trail Blazers when free agency opened.
Parsons has an advocate in Memphis in recently hired assistant coach J.B. Bickerstaff, who formed a close relationship with Parsons during their time together with the Houston Rockets.
Parsons averaged 14.8 points, 4.8 rebounds and 2.6 assists per game during his two-year tenure in Dallas but was only able to play in one playoff game because of knee injuries that prematurely ended both seasons.
The 27-year-old Parsons had hybrid microfracture surgery on the knee in spring of 2015, and the rehab from that operation lasted well into the 2015-16 season. The arthroscopic operation Parsons underwent this spring to address a torn meniscus in the knee wasn't nearly as serious as the previous surgery.
Sources say multiple renowned orthopedic specialists have given Parsons clearance to resume full basketball activities and expressed optimism that his knee would not prevent him from having a long, productive NBA career.
Parsons averaged 13.7 points, 4.7 rebounds and 2.8 assists in 61 games last season, statistics skewed by a slow start as he dealt with strict minutes restrictions. He had the best statistical stretch of his five-year career in the two months before his season ended, averaging 18.9 points, 5.9 rebounds and 3.2 assists per game while shooting 51.9 percent from the floor and 47.7 percent from 3-point range and proving he could thrive at power forward in addition to small forward.
Information from ESPN's Marc Stein and The Undefeated's Marc J. Spears contributed to this report.