The role of the parent is more complex, especially if the parent does not know how to play the musical instrument. Of the three people involved, the parent has the power to create the most change. It is also the most difficult of the three roles, as the parent must assume the role of both student and teacher.
In order to have a young child succeed at a complicated task, a parent must completely internalize the concept that he or she is the at-home teacher. What's more, the child will be having lessons with his or her at-home teacher more often than lessons with a private instructor. Again, heady stuff if the parent does not know how to play.