Facebook is really fun but it can be difficult to convince your parents to let you have one. It's important to remember that they're just worried about your privacy and your safety and how much personal information you're putting out there on the Internet.
There are creepy people on Facebook and they want to make sure that your privacy is as protected as possible. Have a conversation with them about potentially letting you have a Facebook and then convince them with all these tips.
Talk about what would and wouldn't go on your Facebook. Maybe discuss ground rules for your profile pictures and statuses, or talk about what sort of personal information is going to actually be available. It might feel a little controlled, but if your goal is to get on Facebook, even having a slightly controlled access would be great.
Volunteer to give them your password. Honestly, if you can't show your parents what's on your Facebook it shouldn't be on the Internet at all. Because you never know who can see it. The people you think can see your profile are not the only people who can see your profile. Friends of friends of friends. If they're on their friends Facebook or if your friend's friend's friend leaves her Facebook open your information is vulnerable.
If your parents have a Facebook, say sure, you can friend me or give them your password. Letting them see your statuses, and see your Facebook profile itself will make them feel a lot better. My mom is friends with me on Facebook and I don't have her on limited profile, anything like that. Because I just don't put things on Facebook that I wouldn't want everyone to know about.
If you wouldn't say it out loud, why would you make it your status?