I don't agree. An adult doesn't need permission to visit other people because they are responsible the consequences of their decisions. The parents in a situation where their adult son/daughter is living at home can have a say on things that are directly relevant to that, as in don't bring people back, don't have parties, don't waste your money whilst taking all our food etc. Outside of that what you're describing is an unhealthy mechanism to try and keep control over someone.
Healthy parent/child relationships should start as parent/child, but it has to evolve into parent/adult or you start to have a very dysfunctional dynamic. OP paying rent for her own place wouldn't necessarily make her more mature and prone to making good decisions, have a look at the catalogue of absolute morons being posted about in this sub (I'm referring to threads about people with problem housemates, I'm not calling the posters here morons). OP's parent should definitely voice his concerns, I would voice my concerns to my friends too. He has to stop short of actively stopping her from doing something that she had every right to as an adult.
I've seen plenty of examples of these dysfunctional relationships doing real harm to families. I'm in my mid 30's too, not some angsty kid whose parents won't let him go out. My house my rules type of dynamic when used to control someone can cause resentment, as you would expect in an adult. Yes OP is relatively young but definitely old enough where a romantic relationship she has needs to be respected as an adult one.
OP, a lot of people go through what you are going through. Your dad cares about you and people your age do crazy shit. Adults think back to the mistakes they made at your age and try and keep you safe from them. At the same time you're at a critical point in your relationship with your father where he needs to start letting go. Many parents never do. My aunt still fusses about whether I've eaten enough for the day, and my mother was worried about a 2 hour bus trip I was going to take the other day. I haven't lived with them for nearly 2 decades, travelled around many countries, rent several apartments and have a career. With some family members it's never enough, but they do it out of love and I can keep a healthy relationship with them because I don't pander to their insecurities. I've been very blunt with my family members and they still do it. They have never stopped talking to me or loving me just because I tell them "I'm doing X because that's what I think is right for me. Thanks for the advice but it's my call".
Your dad is going to be there for you when you make mistakes, and hopefully you will be there for him because as you grow up you'll notice adults around you also make a lot of mistakes they need help with. Take ownership of your life, and let your dad know he is part of it.