Thanks for the reply and adding context as to your need. As Neil has already said, it is not possible to automatically print multiple parent labels. With your added information I can see why you would want these added labels.
When we were first developing the check-in system, it printed a parent tag for each child. When we originally demoed the system to customers they asked that it only print one so it would not waste so many labels. So if we add this, there will have to be an option somewhere to control this behavior. Do you require this for all classes or just some?