The parent, teacher and child triangle comes up a lot in Suzuki education. The reason why this image is discussed ad nauseum is because there's really a lot of layers to the concept, all of which need to be in place in order for learning to take place.
The first thing to notice is that in a triangle all sides are equal. Successful music lessons are not about the teacher being all-powerful and bestowing the gift of knowledge upon the lowly minions. Every part of the triangle has a job. If any part is missing, the triangle collapses.
The second thing to understand is that music lessons are a give and take experience. Everyone needs to be on the same page. The student will struggle. Struggling is necessary for growth to occur. But it is important that the student knows what is expected of him. It cannot be the parent or teacher doing all the work. The student must contribute as well, even if it is only a grudging contribution.
Finally, the key element that holds all sides together is communication. A teacher can only do so much if the parent fails to voice issues that arise at home. A parent is limited if they do not understand what it is that the teacher expects.