My father was a Museum Curator/Director (now retired), my mother was an assistant curator/Archaeologist (now deceased).
I haven't gone into the same industry as them at all. I think the main way my dad affected me career wise was that:
1) He (along with my tutor) talked me out of changing my degree course after my first semester - I suspect I might have got a higher classification had I switched, and hence got on a graduate scheme.
2) He instilled in me the importance of getting a job (any job) rather than holding out for something I really wanted.
The above may sound slightly bitter but I don't resent him for it, at the end of the day as an adult I make my own choices in life.