You might not have to even have a big talk with your parents about your choice to become a Goth, they might just appcept that this is you and your happy. However, if they believe those lies fed to them by the media about Goth being a cult, all satanists (I'm not saying that no goths are satanists either, and being a satanist doesn't mean that you're evil, it's just not a requirement by any means-not that there are any requirements to becoming a goth) and being depressed you may have to just explain to them about who 'Those scary people in black' really are.
Firstly, don't expect it to be overly difficult. When I told my mum that goth didn't mean depressed, she was imidetly fine with it. It may be just as simple as that. Remember, your parents are just worried about you becoming involved in things like drugs, so if you can convince them that goth isn't a group of druggies and the like, you'll be right on the road to success. Don't get angry, explain things to them in a calm manner, this makes it a lot easier to get your point across.
Obviosly, if you turn from wannabe barbie or whatever to deathrocker in the course of a few days, your parents will be shocked and worried. try starting of with a more parent friendly style of goth, like perky goth (Come on. What parent can resist pink fluffy bats?) or ethergoth (beautiful flowing dresses, calm reputation, what's not to like?) and gradually ease your way into the more showy styles of goth. Also, ask your parents what's ok to wear so you don't scare them and make them clamp down on what they'll let you get away with