So, you don't have to follow my advice, but this is what I would do:
Get them to talk with you and tell you EVERY reason they have for not wanting you to do magic. Carefully listen without arguing with them, and when they're finished go back to your room and write down EVERY reason that they told you. Now that you have their arguments, it's time to "prepare"....
Try to prove their arguments wrong, not in any offensive way of course. By that I don't mean that you should go down and talk with them every time you come up with an argument, write them down instead. When you have arguments/reasons to prove all their reasons wrong or just weird (Once again, try as hard as you can not to be offensive with your arguments), try to come up with reasons for why you SHOULD DO magic.
By this time you only have arguments for why it isn't "wrong" to do magic, now you need to tell them why it's good to do magic. When you believe that you are finished with the writing down part, refine it. Read it through several times and once again make sure it's not offensive.
Now, I would ask for help from those who support you in magic (I guess you have atleast someone, everyone does...). Ask them for tips on arguments and reasons, ask them if they think your text is offensive and if they think you should change anything.
Now, I would recommend you going to your parents saying something like: "We don't have do it now, but I have something that I want to talk with you about. This is something that's really important and really matters to me, so I would like us to sit down for maybe half an hour sometime when you have time and talk about this." If they ask you what you wanna talk about, I would just tell them "We'll go over that when we have the talk later".
What you have done now is making your parents just a little bit concerned over what you are gonna talk about, and they want to make some time to be able to have this talk with you. Another thing they probably will like is that you didn't force it on them, saying something like "I WANNA TALK WITH YOU NOW! NOW I SAY!" and then started crying when they said that they didn't have time. You let them choose the time when they want to talk.
When it's finally time for "the talk", tell them that this is really important for you and start going through your arguments/reasons. DON'T EVER scream at them or sound angry or anything alike that, you should have a very calm and understanding voice, thus not making them angry in any way.
...
This is what I would do, but this may require you being good at finding arguments, which I have understood that not everyone is. If you can't do it alone, I would recommend you just asking for arguments on forums like this one. Why not just post everything that you have written down? I think a lot of us would love to help you refine it.
I really hope that everyone understood everything I wrote, but I know my English isn't the best so you can just tell me if there is any part that you didn't understand and I will try to explain it better. Also, I did NOT say that this it THE way to handle it, I only said that this is how I would handle it. You don't have to follow my advice, and you don't need to agree with me thinking that this is a good way of doing it.
Thank You!
// RedPants