I would classify a pushy parent as anyone who 'pushes' their child to do something they aren't ready or really don't want to do, such as reading at 4. I have friends whose children are reading at 4 and the children wanted to and love doing it. My dd isn't interested at the moment so I am leaving it.
As a parent you have every right to challenge what the teachers think about your child, especially when it comes to what they can do. If you and the teacher are at odds then I would talk to them about why they think your child is only at a certain level and you think they are higher.
I had a few parents (I used to be a teacher) who would challenge me about the level their child was, especially in maths. What I found was in most cases the difference lay in our knowledge of the children's understanding, for example reading is so much more than being able to say the words.
Wanting the best for your child is not pushy. Making your child unhappy, anxious or nervous about achieving is.