I'm so sorry to hear this fr, wow. I'm 19M. Honestly, well I'll try my best to answer the questions you've put forward,
1) Your parents are most likely still seeing you as their 'little girl' that they once had, and, kind of perhaps maybe like, they want it to stay that way for as long as they can make it. that kind of like, 'innocent', girl to them or something idk (like my dad wouldn't allow me to get cornrows he thinks I will look like a thug lol) but I don't understand it though you are 20; most parents want their children to get married off and leave the house asap and be happy with their independence. To see you growing up is not the problem in and of itself but what comes with growing up, essentially, and that is different traits and passions and wants and needs so on and so forth. Though, when yo usay you're a middle child (in my household I'm first and I get all the stick tho lol), is that; do you have any sisters? how do they treat them when they dress if so?
2) Unfortunately I don't know; it seems to be something that has to develop from your parents point of view, but if you can change their mindset through your actions maybe if words won't work with reference to the argument you had with your dad. Actions as in maybe perhaps just, trying, to comply for only a little while and then slowly implementing what you want to do gradually into your life or something... But it's such an annoying thing because it's like, your dad putting restrictions on where you can go or the stuff you can wear etc. I mean, your dad met your mum not by staying at yard nooo it was by going out, but I know for a fact saying that would not be so good. kmt. More on this at the end I'll see.
3) Are you at university/have you been? More help is on hand there. You can stay low in some university that is far away and join societies there which could offer programs for trips during holidays etc. orr, snap is it that you're on summer holiday... damn. Have you got any friends that you could talk to about this situation? it's a really tough one because it's not health for example it's about relationships; and also finance, wow. Because you are 20! An adult, this is your life, not theirs. It's not like they're trying to stop you from doing damaging things such as smoking or gambling this is about clothing and going places like wth.
Honestly, this is such as tough situation try talking to them again. Both of them. Aave a proper sit down talk without feeling emotion and just let what your dad has to say, seeming to be the stricter one, pass by. Maintain your tone of voice as well, keep it calm and collected. Read a summary of the 48 laws of power by robert greene. (full thing will take too much time and not necessary). When I deeped it uni is still quite expensive tho, because sfe doesn't cover everything still unless if you work alongside but again idk you might already be at uni and this is just the holidays. If not though then it means you'll have to then even convince them on moving out to university or to wherever; to do that research up the place highly and talk intellectually about it so that they know what you are on. Let them know that you are only a phone call away+list all the ways you can keep in touch and can come home mid term for a weekend visit and they can come to visit you for a weekend too (againn assuming you're not in uni).
Since they fear when you go out/want to move out to just do what you want/behave/dress anyhow; it is a gradual process. Don't expect them to agree with you overnight, but honesty and hearing out their pov is key. But again I hear that about the argument with your father though but he's said those things he did which may have been hurtful because he does truly love you, as well as your mother as well. I know this isn't the case all the time, but a considerable amount of the time parents act with good intentions even though it may seem controlling/annoying/selfish etc. (my dad not liking the idea of getting braids or dreads or cornrows or earrings is because ik there is a stereotype around people like that with crime/badness i'ma keep a buck) but still, there is always a point where it becomes too much and this looks like its one of them.
Consider in the discussion that yes, you understand that they may just want the best for you. But you are you, and you are your own person and can't live under the facade of being everything your parents want you to be, you're another human being. Outline the benefits for them, too, should you move out (the money aspect I'm not too sure of perhaps see if you've got a trust fund or if not look for jobs you could do,) such as less cooking, washing up, less noise– I know you said to move out asap, but for it to work as in look genuine to them and for you to have money saved I'm afraid it'll have to be after some time. Because if you told them all this right now they would laugh and wouldn't buy it at all. Just comply with them on some things as of for now so that you can start seeing lower restrictions being put in place by not vexing them and even could start removing exisitng ones. It's kind of like playing the devil's advocate here. Friendly disagreement here and there is normal; agree to disagree though they must understand that. Also have them begin to comply with just menial things you want them to get along with. Work up from there. When that time comes outline to them that you won't be the only person getting something out of you leaving.
In the end only you know how to convince them since you know your parents the most. They see you the most, and most likely vice versa, too. So let them see the adult in you. Let them start seeing you being more mature– waking up earlier, getting things organised, writing to-do lists, offering to wash up a couple nights a week, learning how to cook if you don't know how already (I need to do this lol), do your own washing, etc. This will help them massively realise the change that's in you; perhaps if it's not them wanting to see you grow up right before their eyes so quick then it could be perhaps they want you to mature more?? You'll never know with these types of things. Show them that you're ready to go and live on your own. At the end of the day, you are going to move out in the future no matter what– whether it be in a year, or 2 years, or 5 years. Even if they initially dislike the idea because of how they may think of you now try and see if you can implement some of these strategies to try work around that and change their view.
Listen to their reasons but you don't have to hear them out– heck, maybe even read the summary of the art of seduction by robert greene as well (havent read that one tho). They might not be convinced straight away, but I know that they can be convinced eventually. Show them an attitude that you are willing to learn and take care of yourself, which you are and already do but make it more profound and evident to them (stubborn, strict parents we're dealin with) so you can put their minds at ease. It's super super annoyin but it all comes down to just following on for a short while, to gain that 'trust', from them.
I hope this is not too long-winded and repetitive, but um, yeah i hope this can at least somewhat help, and wishing you the best of luck💫🔵