Yogis cannot share their profile with another person to be able to sign up for a class under someone's name.
Each yogi profile is unique by using a name, linked to an email address. The data is thus unique and protected. By sharing a profile the data is less protected. As partners, it is possible to have the same account and use the same membership or class pass. However, the teacher will never be able to see who has signed up for the class. The same goes for a child that is registered on the same profile as the parent.
You could create special memberships or class passes for children that are exclusively used for the classes for children, so you know which child comes since the family name is probably known. A parent can also choose to create their own profile with a new email address under their own management for the child. This is the clearest and safest option for both parties.