I don't remember any of my first days of school. I think it all depends on what happens on that day vs it just being that day. If something really memorable happens, you probably remember it. But if that memorable thing happened on another day, you'd remember it. I think, as other says, that the parents remember it more because it's a big separation for them.
If anything, I remember my first day of orientation for going back to college as a middle-age adult. I had to have my boyfriend with me as I was so nervous. I remember him being there for me.