You are 100 percent entitled to autonomy and privacy.
Some people simply don't like splashing their personal life onto social media or even talking much about their feelings with friends, but that's different from flat-out hiding your partnership, which is about as glaringly red as a flag can get!
We are all insecure on some level, and it's nobody's fault for feeling that way, but no reassurance from a partner will ever be enough, because those issues live inside the person feeling them.
And what does that say about whatever untrusty worthy habits or temptations they might be projecting onto you?
Checking your Facebook inbox in peace without someone mouth-breathing over your shoulder is practically a basic right!
(Unless their parents are abusive, in which case, they're beyond justified in refusing to make nice with them if they so choose.) If they can bear to be super harsh and ungrateful to the people who literally , what does that say about how they'll treat others who get close to them?