He most definitely has ADHD, but because of a bad lifestyle of incessant TV watching. It can be overcome. The lessons are half an hour, though a few times I had to cut them short. His attention span is, however, increasing with every lesson. I think he is acclimating to someone forcing him to focus which is new for him. And you are certainly right: there is NOTHING more exhausting than studying music/playing the piano! The second I ask any student to exert REAL effort of any sort, they begin yawning haha -- not from boredom but because concentration drains one immediately! It happens to me, too, especially when I am trying to memorize (which is very taxing) or when I am analyzing music.

Now, stop this. I'm getting REALLY mad when I read this. You seem to have no idea whatsoever of what ADHD is. (My son has it, my daughter has not.)
First of all: ADHD IS NOT ACQUIRED BY WATCHING TV!!! It has NOTHING, NOTHING to do with your lifestyle. You are born with it. Period! I am so sick&tired of all these "knowing" parents who think my son's behaviour is the result of bad raising, of us parents not being good enough or strict enough or whatever enough we should have been. Truth is, that raising a kid with ADHD is about 10 times harder than raising a so called normal kid. I have both kinds here; I can tell the difference. There are ignorant people enough in this world; may anyone who reads this now be a bit less ignorant, ok?
On the other hand, children with ADHD often watch a lot of TV -
because they have ADHD. They have an inner chaos and the TV helps them calming down a bit.
And from your description, yes, the child behaves just like my son did when he was at that age, so he might have it. Or he might not, it is impossible to tell. But no matter what, I can tell at once that this kid is not ready for piano lessons. His behaviour clearly says that he is too young for this.
Yes, we know the stories about those wonder kids who started playing Rondo Alla Turca before they had even get rid of their dipers and so on, and at the age of 12 they played Rach 3 and yada yada. This is so stupid.
IF the child is ready, yes, go ahead, help him. When I was five, I wanted to learn how to read. I asked my mother about the letters and I learned how to read in no time. Because I was ready! It was my time. But children are different and you are not "smarter" if you learn certain things at an early age, than if you have to wait much longer. Early potatoes are not better than late season potatoes, they both taste good, they are just different.
It is pure insanity to give a 5-year old lessons if he is not ready for it. Tell the parents that he should go and play with his friends instead, that will probably be of much more use to him. He might be ready when he is 10. Or 20. Or 40. Don't take it too seriously when he says "I want to play the piano!" if he at the same time does everything to stop you from teaching. He probably wants to play that he is playing, he does not want to learn for real. Not yet.
So tell the parents that this child needs no lessons right now. Expose him to the environment, let him hear music, sing with him, let him watch other people play, let him sit at the piano and do whatever he likes there (as long as he is not damaging the piano, of course), and the day he really IS ready, he will learn quickly and have a good foundation to build upon. Or force him to have lessons that will make the teacher irritated and bring HIM to tears as he feels the frustration AND the teacher's (don't imagine for a moment that he does not sense this) and you have prepared him well for a life with bad self esteem and a solid hatred for playing the piano.