It actually leads to, at least, a theoretically interesting situation where parents could authorize almost any adult they left their child in the care of (such as a Scout leader, sports coach, Youth club leader, Nurse, or babysitter, etc) to administer corporal punishment, with state school teachers being the sole exception to that rule.
(Teachers in private schools remained under the primary consideration of in loco parentis under these court cases – the Crown powers argument only applied to teachers in state schools).
The main court case on this, by the way, is Ramsey versus Larsen (1964). That case did not directly involve corporal punishment (rather, it dealt with negligence and accidental injury when a boy fell out of a tree) but is the case that said teachers in state schools derived their powers from the Crown, not directly from parental delegation.