Should I put a sign on my office door saying what days I work? Or is it better not to, to create the impression that I'm always around?
You shouldn't have to hide the fact that you work part time. That's all you get paid for! You ought to have the right to bring your whole self to work; you the parent and you the professional. Perhaps rather than on your office door, you could just leave the information somewhere inside your office - on a keyboard or on your computer screen if visible. Don't shout it but don't hide it!!