Well, that's not my experience. One you fail you fail. Could you fail mid-rep? Sure. But falling to the ground from 2 feet up isn't going to injure anyone. In any case, pushups have been used as punishment by the military, by coaches, etc. since time began. I doubt that anyone's ever been hurt doing pushups to the point of muscle failure. If that guy was able to have his boys do pushups for 30 mintues straight, they now have the potential to be Olympic champions; that's an extraordinary feat.
A 30 minute period for one exercise suggests to me multiple sets of working to near failure and with each subsequent set, the cushion between an active muscle failing without injury decreases. That's taking into consideration that the proper form is used, which rapidly falls apart the closer you get to absolute failure. Then you have to take into consideration ligament and joint problems. Rotator cuff injuries are much more likely than a grade 2 muscle strain in this case but ligament and joint issues are a lot tougher to overcome and heal from than purely muscular ones. A muscle injury is still possible though. I'd be interested to know if previous untreated injuries have damaged their full range of motion and overall strength, negating the positive effects for fitness to begin with.
Remember, this was a stated
punishment. Presuming the father is only using exercise as a punishment and not as a healthy routine it places the risk for injury much higher. At 5'6 and 320 lbs, I'm not sure he has any working knowledge of basic fitness. For the sake of his children I hope there isn't any lasting damage.