Yeates, 30, will officially leave the Coral Windows Stadium when his current Bantams deal expires at the end of this month.
The former Watford, Sheffield United and Colchester United attacker made 81 appearances for City, scoring seven goals in the process.
Parkinson said: "I sat down with Julian (Rhodes) last week after the take-over didn't happen.
"We looked again at the budget for next season and the players we brought in and unfortunately we decided we couldn't offer Yeatesy a new deal."
Yeates' departure comes following similar exits from the club for Matthew Dolan, Jason Kennedy, Andy Halliday, Matt Urwin and Aaron Mclean.