I am 64 and have lived on my own for the last 14 years. I am practically living in my bedroom, because every time I am with my mother, she picks a fight, them blames me because my dad gets very upset. Things are just terrible, everyone is depressed and I don't know what to do.
I have more patience and can better handle the emergencies that arise now that I no longer have my very selfish mother in my house. You have to take care of yourself because if you burn out, then there's no way you can help them. Help from a distance (like another house) is better than no help at all if you have a complete nervous breakdown.
Importantly, THEN you will be able to help. When you're not spending every day deflecting your mother's anger and mopping up your father's distress, you will have a great deal more mental space and physical energy to devote to getting them practical support. But first of all, stop taking the bullets.
As an aside, your mother probably does feel genuine rage. No doubt because I'm not on the receiving end of it, I actually feel very sorry for her. She must be frightened, nothing she can do will snap your father out of his dementia and concentrate his mind on her again, the future must look very bleak to her… she has things to be angry about. None of them your fault, of course; but I think it helps sometimes to see the "why" of things.
You intended the move, I'm assuming, to be helpful to your parents. But it doesn't sound like your presence is helping (and that is Not Your Fault). I am not suggesting abandoning them. Advocate for them. Help them get the help they need. Pick Dad up and take him on outings. Go to doctor appointments with him and understand what is going on with him. Just don't do these things living in a bedroom in their home.
Continue to love them, but detach from your mother's toxic environment.