Phoebe, sensing how close to the edge Quinn actually was, didn't let him call his parents right away.
She instead prioritized getting food into Quinn, hydrating both of the crying men in her life, and making a battle plan that Quinn could follow. She knew her Aunt, and knew that he would need it when something like this was happening.
The first thing they did was let Quinn's parents know that he was going to call them in an hour. Giving them a heads up was a smart plan, because what if they were busy? Quinn was the one who insisted on that, even if Phoebe thought he was being ridiculous.
She never gave her parents a heads up, but Quinn was on the calmer side of hysterical, so she listened to him. His Mom sent back a question mark, but said they'd be ready for the phone call. That was one thing out of the way.
