I too refer people to other teachers I know. I take their name and number and tell them I will call them when I have something open, but they ought to look for another teacher while the child has interest (or while the parent has interest).
I don't think you ought to worry as much about wasted time for a year or two if that's what the parent chooses to do. You can only do so much, like encourage them to find another teacher, and honor their decision if they choose not to.
However, there is plenty that a parent can do in that time off; things that have nothing to do with YOU. A motivated parent who thinks music is important enough to get on your waiting list will find ways to involve general music in their child's life. Here's a few things a parent can do to keep music in the child's life while they wait for an opening in your studio:
* Take children to concerts -- orchestra concerts, band concerts, solo concerts. Universities have tons and tons of free student recitals so parents don't even need to invest a lot of money (if they don't have it to spend on concerts)
* Read children's books about music -- this can help pique childrens' interest in the subject matter
* Bring children to YOUR studio recitals even though they aren't yet taking from you -- this gives them an opportunity to get to know you and see other children playing the instrument. It also gives you a better indication if the family is committed to the idea of lessons and all it entails.
* Play CD's of quality piano music (or any classical music) at certain times of the day -- getting up in the morning, during breakfast, during study time, at bedtime. The parent can easily ask kids questions about the music such as: Was that a high or low song? Fast or slow? What kind of emotion did it sound like? Happy, sad, angry, playful, etc. That at least gets the child thinking about some of the less technical aspects of music, but he's still thinking about music.
* Easy music theory software -- many of the software packages that teach basic music theory are easy enough that a parent can most likely follow along as the program does instructing and help the child understand the goals of each "unit". Many of these elementary school theory programs even let them create their own music, and does some note-reading teaching too. In a quick Google search I found a page with a list of software solutions:
https://magicref.tripod.com/articles/musicsoftware.htm I'm sure you could find something to suggest to the waiting parents from this list or your own search that encorporates some of the things you teach in lessons.
* Join a choir -- we all know singing improves how well kids learn and perform music on any instrument. Even if it's something as basic (and free) as participating in a church children's choir, the child would be involved in music, learning about dynamics, phrasing, and maybe (MAYBE) note reading. The old cliche about singers not reading music is often sadly true.
I'm sure there are lots of other ways the parent's can keep music going in the child's life and maybe you can even think of a few other things. All of the above avoid having the child try to actually play piano and practice, which could build bad form and habits before they have actual teacher guidance. If they want to play so badly that they can't wait, then they really ought to take from another teacher to avoid forming those bad habits. It's just too hard to undo them later. A final thought: learning patience and how to wait for a good thing isn't necessarily a waste of time.
Best of luck!
Astyron