"My parents thought that you were our teacher."
Translation:Мои родители думали, что вы наш учитель.
Here the English is ambiguous. "My parents thought that you were our teacher" could mean
a) they thought you were currently our teacher b) they thought you are the teacher that we used to have. i.e. they thought they recognised you but got it wrong**
Am I right in thinking that Russian does distinguish between the two interpretations, and would use был(а) in the latter case?
Yes, but if you are speaking to someone older, in a position of authority, or a stranger, then you should probably use вы