I am much older and was once in your situation. I grew up and really understood what it meant to be an anarchist. And is has not much to do with the "obey" but rather with the "obey blindly and accept it".
I believe the fact that they do not trust you with a computer arise from the fact that you are not mature enough — or at least the fact that you don't show them enough maturity. And I think, by the tone of your post, that you are not mature enough.
That is not a bad thing, at all! You need to take your time to grow up. And if you want to have more rights, in their home, try to understand their side of things. Ask them why they won't trust you, where does the need to restrict you come, how can you act to have more freedom.
Anarchy isn't about blindly disobeing. That's just disobeissance. It's about understanding what rights you don't have and the reason behind it.
One last thing : please don't forget that even if you think you are ready for the outside world, at 15 you are not. Some fifteen years old are, but they are not the ones to come on an internet forum to sort things out — they already have it all figured out. For the while, communicate with your parents and try to understand their point of view. And remember that as long as you live under their roof, you have to abide by their rules. You can also choose to continue breaking them as you are doing right now. But be smart about it.