Houston will now have to decide whether to match the offer.
Several photos of Parsons signing an offer sheet were posted on a Cyber Dust account controlled by Mavericks owner Mark Cuban and the team early Thursday, when the NBA's moratorium on signing free agents ended. The photos, including one of Cuban posing with Parsons, appeared briefly just after midnight Dallas time.
The Rockets have three days to match the offer, though it wasn't immediately clear if they would do so for the 25-year-old forward who has played for them the past three seasons. Parsons was a second-round pick out of Florida in the 2011 draft.
Dallas got an agreement in place with Parsons even before small forwards Carmelo Anthony or LeBron James announced where they will play, a signal that the Mavericks are most likely out on the two most prominent free agents.
Anthony visited the Mavericks at the start of the free agency period, but Dallas owner Mark Cuban has acknowledged that he couldn't offer the forward a max contract.
Parsons averaged 16.6 points, 5.5 rebounds and 4.0 assists last season for the Rockets. He has played 213 games, starting 207 of them, while averaging 14.1 points, 5.2 rebounds and 4.1 assists.
In Dallas, he would be in a frontcourt with Dirk Nowitzki and recently reacquired Tyson Chandler.
Nowitzki and point guard Devin Harris have already agreed on new contracts that will be able to be signed as early as Thursday. Chandler, part of the Mavericks' only NBA title in 2011, rejoined Dallas in a trade from the New York Knicks.
Shawn Marion, the primary small forward for Dallas the past five seasons, is a defensive specialist and unrestricted free agent.
AP Sports Writer Schuyler Dixon contributed to this report.