Maybe there is a philosophical divide between us, but this argument sounds really weird to me.
From his/her point of view, of course being born to educated parents is luck, if not what else is it? We don't have any choice or any influence on the matter of who we are born to, and we cannot change it with our effort or lack of it.
The fact that from their parents' point of view it's not luck, because they worked for it, strikes me as irrelevant in this context. It can be relevant in a societal context but the post you reply to was talking about the success of an individual and how it can come just by luck, without that individual working especially hard for it.