I agree only a little with the above advise. Talking with your mom and dad is always a healthy step. Realizing changing their diet would be a major life change as well, you need to control what you can...
1. The amount you eat. When you are losing the battle with the 'tempting snack' get a small bowl or baggie so you can limit the amount your eating. At Neal times only take a single serving. Once you are done eating clear your plate so you are not tempted to take seconds.
2. Stay out of the kitchen. The more often you go where the food is, the more you will be tempted.
3. Exrersize. You don't need to go all out. This only sets yourself up for failure. Start small, walk, jog, an easy exersize video, anything you can add to your day. Do it regularly. After a little your body will want more. Increase the intensity of your workout a little at a time.
I hope this helps. (And on a side note, giving your parents the ultimatum that you will move out is bad advise, in my opinion. I am sure they love you and probable feel or have felt these same health struggles you are now and don't have the tools to overcome them.)
You my sympathy, your situation sounds hard. You need to be practical and either try explaining to you mum you need healthy food in the house to eat properly or consider living somewhere else. If you have grandparents still alive, they might be more supportive. If your mum realises how serious you are about eating better, she might be more cooperative.