If I were in your shoes, which in a way I am, I would try to get father to go with me to a lawyer and get a Durable Power of Attorney, I know your parents do not want to give up each other, even if one is crankier than the other. They have a lot of memories in their minds, even if some dementia is developing. Let Dad know you will help out and do their bookkeeping etc. for them. Help with Dr. visits and so on, help mow the grass or sell the house to move to smaller quarters, so your mother won't feel so overwhelmed.
These things will take a lot of you and your brothers time, but it will all be worth it, you will have no guilt to come back on later. You don't need your mothers permission, to do this for your dad, explain to him how if something were to happen at this stage, the law would force them to be put in homes, maybe not even the same one.
I take care of my 83 year old mother she has had a stroke and developing dementia for 5 yrs at least. She is grumpy and at times unbearable. So far with POA, I have helped her sell her house(she could not live alone), got just about enough money to have her teeth fixed and help pay for the burial that she wishes for. No help from 2 siblings(they want her in nursing home). She knows who we all are, but as some old people do is jealous of the grandbabies and the attention I give them occasionally. Just as you say your mother is about home health aides, with your father. I've a long tradition in my family of respecting my father and mother, and try to keep Mom as happy as possible, Dad died about 20 yrs. ago(heart disease). Please keep trying with your Dad he will probably go first, but see if he understands enough about taking care of Mom when he is no longer there. Shute, make up a story about where you are going, (when you go to a lawyer) with Dad, about going out for ice cream or to get a surprise flowers for Mom. But you do need the POA and Advanced Directive or Living Will just in case something unforseen happens.