Should a girl continue Dawah work at the university even if her parents disapprove, out of their fear of security issues, while she believes their fear is unfounded? And would the answer differ if the questioner were a man?

Obedience to parents is obligatory in matters that do not involve disobedience to Allah. If the parents forbid their daughter from da'wah activities out of fear for her, then she must obey them. She has a wide scope for da'wah in other matters about which they do not fear for her.