Thank you for your question. I look forward to working with you to provide you the information you are seeking for educational purposes only.
Generally, when a parent takes issues with multiple programs and teachers at the school and then try to claim, "history suggests the problem is not with" the child, they are lying to the school and to themselves. The problem here is with the child and as a teacher who is now seeing that this parent is complaining about not just you, but the prior teachers at the school, you need to contact your administrators about getting the child tested for whatever special needs that the child may have so that the school could address them.
You should NOT meet with this parent alone without the administrators present. This is a parent who is in denial about his child and is bent on blaming everyone by the child for the problems the child is having and instead of taking action to get the help the child requires, he is blaming the teachers as his angel is doing nothing wrong and she is a regular Einstein genius.
You need to also point out to your supervisor that if he reads the email he is not complaining only about your class, but it seems like every other class this student has been in has been a problem as well and that the problem is with the parent and student and not the classes and you need to convince your supervisor and administration to get the child tested so her needs can be met.
As far as a parent complaining about you, I am afraid that legally there is nothing you can do, it is the nature of the job you have undertaken and as you can prove not only you did nothing wrong, but this parent has a history of complaining about his little genius child and blaming everyone for any failures, you need to point this out to your administration to protect yourself. You should write an account of what happened that day and deny it and then tell them they can ask the children as well to verify you did not ask her any questions that day as alleged in the email so the administration knows this parent is just plain trouble.