Oh yes, the parents who 'helpfully' write "this book was too easy" as the comment in the reading record EVERY. SINGLE. WEEK, then when I sent home an extra reading book which came from a different scheme, they didn't bother reading it, as it wasn't ORT, so they didn't realise they needed to read that one, too, apparently, despite the fact that I'd written it in his comment book. I'd class that as irritating, rather than actually pushy, though.
Some exaples of what I would call 'pushy' parents:
*The ones who steal ORT books from higher levels/ further along the level their child is on, so that they can help the child memorize read it at home. This is designed to make it appear to the teacher that the child can magically read the whole book fluently the first time they are given it in class, but is fairly easily spotted by the fact that they can also read the words on the following page without actually seeing them if they don't turn the page over in time!
- The ones who do all homework for the child, instead of with them, sometimes to the extent that if you talk to the child about it, they're not actually sure what the work is that they're supposed to have done.
*The ones who, upon learning that our new topic will be Victorians/dinosaurs/space/whatever, spend the entire weekend cramming their child with facts on the topic, so that they already know 'all about it' by the time we actually start the new topic. Far from making them more interested, it can actually make them a bit bored with the topic, as they don't get to find out new things like the rest of the class do.
*The ones who are generally obsessed with making sure that their child is 'the best' at everything, even if they child isn't necessarily able to reach that standard academically.
There's a world of difference, though, between being interested and being 'pushy.' Taking an interest is a good thing, helping your child to do their homework, pursuing interests they do have with them, at their own pace, speaking to the teacher if you feel a specific problem has arisen- that's very different to being pushy, IMO.