Wow...I really feel for you. Not that I know what you're going through, but hearing about it just makes me so sad. It puts a burden on my heart, and I'm praying. I just encourage you to remember that your parents have the best of intentions, and I'm sure they really care about you, even if they are wrong sometimes. I definitely agree with most of what seriouslycoolwife said--you want them to realize that your desire comes from a pure heart that is really seeking God, not just wanting to have your own teenage way. Allow God to work in you, submit your will to His, and to your parents where they agree with what He says...
I would not go so far as to say, though, that you do not need a church. God created community for a reason, and we find it perfectly manifested in the essence of His being, the Holy Trinity. I shudder to think of where I would be without the blessing of the church I attend. I know some people go to a church that is not as helpful to them, so it is less valuable. But that does not lessen our need for fulfillment, and also to be a blessing to others. As far as your parents' religion, I know that I could never be content with Catholicism either, though, it would just feel so wrong, unfulfilling, old testament. Don't get me wrong, I believe there are many sincere Catholics who will get to heaven, but not the ones that rely on the sacraments, if even partially, for their salvation.
I will be praying for you. May God give you the faith in Him that alone can lead us a life of fulfillment with the Source of all Life, Love, and Light, and the knowledge to know how to do this, and the strength by His Spirit to be able to by His Power.