There is no definite answer everybody agrees as to who is the client in education. All these could be considered clients: students, parents and society. But I believe that our main client should be the student.
Maybe it comes from my marketing undergraduate degree, but the client to me means "the target". If we are targeting the student, we are on the right track.
Targeting the parents is not always a good idea. Their interests could be focused on their children's grades, schedules, money, and other aspects that don't necessarily have to to with student learning. It is also very different if we are dealing with a public or a priviate school. That is why I think understanding the student as the client is the silver bullet.
My main goal is to make students better learners. Life-long, responsible, critical-thinking, self-motivated learners. That is why it is it so important to teach competencies integrated in the curriculum, such as environmental and social responsibility. If I succeed, then students are going to give back to the community and create a ripple effect. So at the end, you are contributing to society as a whole.