Pushy parents are terrible. Skipping levels in piano for a student who is not ready is like skipping a C or B student a grade at school. It's not in the student's interest, but rather in the prestige in the eyes of the parents. Some of the parents of my students say "why is so and so not doing their grade 1 exam yet", and I reply "because they are not ready; they are not mature enough to understand the reason behind PRACTICE, they forget their books, cannot locate middle C every odd week, etc". These parents are lazy at home, but like to push the students during the lesson, or in front of the teacher.
It's really good to stand your ground - usually (in the sense that the parent has no musical background) you will be right, and will actually have to explain why the pushy parent is NOT right. I do have trouble doing this, especially when it's the husband and wife who are both there standing THEIR own ground, in front of several other parents. Just make sure you know that what you are saying makes sense, and reason with the parents.