The Rockets have until 11:59 p.m. ET on Sunday to decide whether to exercise their right to match the three-year, $46 million offer sheet Parsons signed with the Dallas Mavericks earlier this week.
Source close to the situation told ESPN.com's Marc Stein that the Rockets, who can exceed the cap to keep Parsons, are still weighing whether to match the offer.
USA Today earlier reported Ariza's deal with the Rockets.
The Rockets must decide whether it's worth paying Parsons more than $15 million per season over the course of the contract without accomplishing their goal of adding an All-Star power forward this summer.
Ariza would have been on the short list of small forwards the Mavs wanted to target if Houston kept Parsons.
The Mavs are maintaining contact with the agents of free agents Luol Deng, Lance Stephenson, Paul Pierce and Shawn Marion, who played the last five seasons in Dallas.
The Parsons offer sheet, which starts with a $14.7 million salary, accounts for the rest of the Mavs' cap space after the soon-to-be-official deals for Dirk Nowitzki and Devin Harris are done.