How do you address the issue of parents who clearly need support but refuse it?

My mother has been diagnosed with moderate-stage Alzheimer's and is completely unable to make rational and logical decisions. My father is nine years into a Parkinson's diagnosis, and although he has serious physical limitations, he is still "legally" cognitively healthy enough to act as my mother's caregiver and decision-maker (although I have a strong suspicion that his executive functioning and ability to reason are somewhat impaired by the Parkinson's). However, he has always dealt with conflict by avoiding it. Because of this, he refuses to support me and my brother's attempts to provide much-needed in-home support because it upsets my mother and causes her to be very difficult to deal with. We tried making meals for them, but they wouldn't eat them and continue to eat most meals out at fast food restaurants. They rarely do laundry and we worry that even if they do, it requires my father to walk down steep stairs to the basement. My mother has always had some issues with hoarding and their house is full of clutter and fall hazards, but they refuse help cleaning it and don't allow us inside without serious conflict. My mother has convinced my father that we are trying to kick them out of their house and into a home, despite the fact that our attempts to provide support are precisely so they can stay in their home as long as possible. They will not allow anyone in their home, so we're not able to get my dad any respite from caregiving, unless it's provided by me. My father refuses to sign POA, advanced directives, etc., because he is certain we will try to take over his decision-making and autonomy. But, because my father is not at the point at which he would be deemed incompetent, our hands are completely tied. Our family has always been close and gotten along well and this has completely destroyed our good relationships with our parents. They are convinced we are the enemy and it's us against them, even though we've not forced the issue at all, only tried to work with them to find solutions. We can never talk to my father privately, as my mother is always there and he immediately puts us on speaker phone as soon as we call, so all conversations are three-way with someone who has memory, cognition and mood issues. My dad continues to allow my mother to be the "point person," which leaves her getting the brunt of our frustration, when she can't even remember what was said at the beginning of conversations, much less in previous conversations. They've begun not answering my brother's phone calls and I fear that they will start doing the same with me. We're looking into working with an elder care attorney and aging care social worker, but have no idea where to begin and don't know what we can really accomplish while my dad refuses to work with us. Any advice about next steps would be greatly appreciated!! I'd also love to hear any ideas that would help me lower my frustration level. I've been trying to make progress for three years now and we've gotten nowhere, so I'm to the point where I practically erupt during any kind of serious conversation with them, which I know is not helpful. I have a three and five year old at home, work part-time and am expecting our third baby in March and so feel completely overwhelmed, and to top it off my father is scheduled for another medical review of his driver's license in April (he lost it temporarily two years ago and it was a disaster!).