Is it permissible for me to receive Social Security disability benefits if my father forged a document proving his poverty, even though we are not poor, and out of fear of my parents' anger?

You should gently advise your father and remind him of the prohibition of cheating and lying. If he responds, then that is good, but if he does not, then you have fulfilled your duty. Do not submit the papers yourself. If you are able to stop it and inform the responsible authority without harm to yourself, then do so. Otherwise, the sin falls upon your father, not upon you. As for your expenses, they are obligatory upon him from his wealth, and there is no blame upon you for taking from his mixed wealth.