The Seahawks do not expect Parkinson's injury to be season-ending, the source said.
The NFL Network first reported Jones' injury, calling it a Jones fracture that he had surgically repaired on June 2. The NFL's transaction wire for June 2 noted that Parkinson signed his rookie contract.
A Jones fracture occurs in the fifth metatarsal of the foot and almost always requires surgery.
Seattle drafted Parkinson in the fourth round out of Stanford, where he caught 48 passes for 589 yards and a touchdown as a junior last season. He joined a crowded group of tight ends that includes free-agent addition Greg Olsen, plus Jacob Hollister, Will Dissly and Luke Willson.
Dissly is coming off a torn Achilles -- his second major injury in as many NFL seasons -- and may not be ready by the time training camp begins. Olsen, 35, missed a combined 18 games over the past three seasons. Willson is competing for a roster spot along with rookies Stephen Sullivan (seventh round), Dominick-Wood Anderson (undrafted) and Tyler Mabry (undrafted).