Hey Ryan,
I agree with Kate. It sounds like the problem is caused by your parent's relationship with each other. If it is possible, could you move out and live on your own or with a friend?
If not, then could you at least speak to your parents about this, if you haven't already. You could sit them down together or separately (whichever is better according to you) when they are in a good mood and explain to them that the other person is still your parent and you need a relationship with them as well. In case of her mother, you could inquire about her concerns and give her a chance to calmly explain her feelings as well and then try to put her concerns to rest. You could then come up with a mutual agreement on the matter. You can try to explain your need for a father in your life and ask her if you can spend few hours with him in a month without it affecting your relationship with her.
Let them both know that you value each of their presence in your life and how it affects you when either of them is angry or absent.
You could write them a letter or an e-mail if you are more comfortable with that. A letter will also give you the chance to get across all your points without interruption. Though it is important that you ask them on their views and the problems that they have with the situation and have a face to face conversation once they have read the letter.
Here is a list of hotlines that can offer you more support and advice-
http://www.teenhelp.org/hotlines/
I hope that this helps.
Please feel free to get in touch anytime.
Stay strong and all the best!