I was in a similar situation for almost 18 years so I can relate to you better. Here's my story of how I turned my overprotective parents who mostly caged me in the house, to free bird .
You can't convince them in just one day. It takes time, depending on how overprotective your parents are. This world wasn't made in a single day. So take some time to make them realise that you are grown up and not a kid. I assure you that having patience here is worth it.
If you have overprotective parents, you'll be familiar with the fight that comes with your wanting to go out of the house or hanging out with friends. Depending on how overprotective they are (there's definitely different levels), you may or may not be in for a real fight, even if you're just wanting to go out to see a movie with a bunch of friends.
For me, my dad is always super-protective, and there is a simple 'no' from his side with no further discussion. My mom however disliked this but couldn't take a stand against him.
When I was younger I couldn't stand against my dad, so I grew up never hanging out with friends or leaving the house.
Throughout the school and middle school whenever someone asked if I wanted to do something with them, I would ask my mom first and her reply would be "Ask your dad" and eventually be guilted into not going.
After that, I stopped trying and my friends stopped asking if I could do something (as I never could).
Keep in Mind...
Prepare Yourself and Your Resolve
If you've reached the point where you are tired of not being able to go out with friends, or you want more freedom for yourself, the first thing you need to do is ready your mind. This is crucial. The first time you make your stand against your parent(s), you'll have to fight against caving in.
For me, this didn't happen until my 12th. Yes, I know that is a very, very long time. I regret not going to any of parties with my friends or even movies.
Although my parents were so protective that I wasn't allowed to even play outdoor games. And as a result I am very poor in outdoor activities.
I did really want to go to the party but my dad somehow ended up guilting me into not going. I ended up crying and just thinking "fine, I didn't want to go anyway."
Tips to Keep in Mind
If you're parents know your friends/who you'll be hanging out with, they'll be more likely to say yes.
Don't push too far, too fast. Take it slow.
If they say no right away, try bartering with how good you're doing at school and how you'll be back before a certain time.
Keep in touch with your parents and let them know you're okay when you're out. (It's annoying, but a small sacrifice)
How Make Your Stand
Before even confronting your parent(s), keep your end goal in mind. If it's to go out with friends to a movie, don't let them win right away. It's okay if you don't get what you want at first. Try to barter and keep it at a level that they will be comfortable with.
If your parents don't want you hanging around with the opposite sex (which is another issue, but relevant), assure them that there will be a lot of people going. If they don't like you being out past a certain hour, see a movie and be back before that time. After a while, you can start asking for more. They will get more comfortable with you leaving the house. Just give them what they want somewhat (like a text message when you get there) and don't push too far too quickly.
Honestly, it's a lot like taking baby steps. Don't expect them to be okay with you having a serious date and not coming home until 1am the first time you ask. Let them build their trust in you first and get comfortable before stretching their limits.
And really, it's not so much as "asking" as putting your foot down. This may be hard at first, but if you open yourself up to a "yes or no" question, they're going to automatically say no, because they said so. An example of this would be "I want to go to the movies this Friday with so and so. A lot of people will be there and I'll be home by xxx. I'll text you when I get there."
Depending on how they answer, you may end up having to barter and convince them. "I've been doing really well in school lately and I don't go out a lot. I'll be sure to be careful, xxx is driving." It also helps if your parents know who you are hanging out with. If you have to bring them by, go for it. Play video games, ask if they can come over for dinner, etc. If you're parents are comfortable with your choice in friends, they'll be less likely to say no.
If you give up, you'll always regret watching your life pass without you doing anything about it.
Be Patient and Keep Trying
I can't stress how important it is to keep trying and pushing your boundaries. Depending on how overprotective and how hard it is for you to get your parents to agree to let you have your way, the longer it'll take to get them used to the idea.
One of the things my dad always complained about it that I was "given a little freedom and then I took advantage of it." Yeah, because I had been cooped up for years and years! But realistically, this made it harder to get him used to the idea of me going out. I probably should have taken it slower, but by that time I was 19 or 20 and I knew that he was being unreasonable.
Be prepared for "I'm not talking to you" and the silent treatment every once in a while. I luckily had my mom to help calm my dad down sometimes. Eventually though, he did finally get used to me leaving the house, hanging out with friends. It wasn't in any way easy for him to accept it, but I tried to make it easier by spending time with him.
It's not easy, but if it's something that you want, keep trying. It's well worth the struggle and once you finally do gain your freedom and independence, you'll find that you don't feel as anxious, lonely and caged up as you used to.