Girlfriend not updating relationship status

I have a long distance relationship with her. I met her on fb . she said she wants to marry me . But doesn't update her relationship status on her fb .Her profile still says " single". She gets really jealous though when I talk to other girls and tries to tell them that i am taken. Whenever she sees another girl hitting on me then she comments on my fb pics "are u cheating on me ". She even mentions in fb groups that she is my girlfriend. When I asked her two weeks later after i met her on fb she said she can't update her relationship status or even hide that she is single on her fb because I haven't gone out on a date with her in real. She has introduced me to her friends and classmates but I haven't talked to her family yet even after almost 4 months. When I asked her again after 3 three months since I met her, she said she talked to her parents about me and her parents don't allow that is why she didn't update her relationship status. I forgot to ask her to hide her single relationship status this time though. My confusion is that even if her parents didn't allow she could still have hidden her single relationship status on fb. Is she not ready for a relationship yet or is she seeing other people.
What should I do . break it off with her or continue as it is or ask her again to either hide or update her relationship status