Thank you. That makes it difficult (for you)...since if your dad is not competent, he is not able to make decisions that affect himself (decisions that can impact his person...his health, and his estate
If he is not competent? He truly needs someone to make these decisions for him.
So lets get on to your question
Am I legally allowed to deduct from dad’s account a reasonable salary for myself?
The answer is no.
You are not. Not legally.
Not without a court order.
If you did so, you would open yourself up to charges of fraud...either in civil court (perhaps from your brother) and conceivably in criminal court.
Under the law, when a person has a the responsibility of trustee, they have a fiduciary duty to the estate. That duty requires they place the interests of the estate over their own interests. If you pay yourself, you are setting yourself up for a charge of violation of your fiduciary duty.
Now...clearly your father needs care. And there is no reason you should be working for free. What you may want to consider doing is to go to court and have yourself appointed as guardian to your father. At that point, if you are appointed guardian, you would be able to take steps on behalf of your father to protect his health and his estate. And paying for health care would be one of the items you would have power to do (in he role of guardian).
But what you describe, since you are not in the position as court appointed guardian and since you do not have power of attorney, you do not have this ability.
Sorry to have to bear bad news
Let me know if you have more questions...happy to assist if I can