My house, my rules and until they are out the door everyone needs to follow them ... so yes if a child is misbehaving during drop off or pick up and the parent does not quickly address it in the manner expected in our code of conduct that YES I would step in because I do not want negative behavior at pick up or drop off escalating to other children who are watching and think 'hey maybe I will try that look at all the attention Johnny is getting with that fit!
A child who HIT their parent would get a very stern talking to by me specially if their parent tried to be all 'oh honey' about it .... NOT ACCEPTABLE regardless of the age of a child to hit another person!
A child who cannot not 'exit the program gracefully with dignity and respect for others' the parent is basically instructed by me to 'time to take them as is please and leave because this is not acceptable behavior for the other children to be watching and should not be negotiated with' ... I had one child leave my home in the middle of winter with no coat or boots cause she was refusing to get dressed for the mother she NEVER did it again because it was bloody cold outside and she had to ride all the way home COLD. If the first time did not solve the problem than from that point onward they would be required to tidy up their activity 10 minutes before their parent 'normally arrives' and to get dressed to go home and be sitting waiting the next day on the step all ready to be picked up ~ this way they can be taken out immediately if they were 'acting up' again and they would continue to do that daily until the child has left the program 'gracefully with dignity and respect' and then they would be allowed to earn back the privilege to continue to play up until their parent arrived ... it usually only takes 2-3 days for the child to realize it is 'easier to behave with dignity and respect' because the tantrum because they do not want to go home does not work!