While it might not be the type of experience that they were expecting, this is exactly customer relations!
Experience dealing with the general public, one-on-one
Customer service and communication skills
This type of experience is different to the experience with tutoring, the student-tutor relationship will likely be of less value in this role, though it is still important to include on your CV.
The important aspect is your experience communicating with the parents, they are the customers in the tutor engagement, so focus on your ability to communicate and mediate or negotiate with the parents. In fact parents of children being tutored (in my experience) often provide the most character building interactions you might face, their interest is always deeply personal and so this can lead to some interesting exchanges.
I would suggest something along these lines:
I have been working with XXXX for XXXX years providing free / low-fee tutorial classes to the children. This was an offering to the general public so there was a mix of new customers I had not been introduced to an important aspect of the role is to develop these customers into long term clients. Apart from the initial interactions there was frequent 1 on 1 interactions with the parents where we would discuss expectations, progress and strategies to continue study outside of our session times. I feel my experience will help me to establish good working relationships with customers, especially when they have issues that they are passionate about and expecting answers or solutions. I respect the fact that everyone has their own personal priorities and points of view and my experience has helped me develop strategies to diffuse volatile situations if they occur. I am by nature a problem solver and enjoy the challenge of identifying the real issues and presenting practical solutions to clients. I have been able to build a good rapport with all of my customers to date.
Insert a mention of a reference if you can, one of your "customers", or describe or be ready to describe an actual scenario where you were forced to think on your toes and deal with a situation that could easily have escalated out of hand.
See how we've easily turned a might not be appropriate into the perfect candidate for the role ;)