M.P.
I think your Dad is remembering how it was when he was a youth. My aunt did this in the 50's and 60's and it worked well. The world is a different place now. It's impossible to find a live in caregiver on your own and even if you did you have no guarantees that this person will be helpful let alone honest and hard working.
Your mother may be eligible for help in paying for a caretaker thru Medicare and her insurance plan. I'd talk with the social worker in her doctor's office about what might be available. You do have to hire this person thru an agency in order to qualify for financial help.
I've had relatives and a neighbor have various people in to help with various tasks and had all of their wages paid by Medicare and insurance. Once they needed round the clock care they had to rely on a relative moving in with them or them moving in with a relative or to a care facility.
My father moved to a foster home. My mother could've moved with him but chose not to. My brother moved in with her. Foster homes are more personal and less expensive. My father was glad to live there. He, as well as family members, became friends with her. My mother and brother could visit anytime they wanted within reason. The caretaker became an advocate for him with the family and his doctor.
A friends father had a similar experience. I see this as a viable option for older people, including me, when I need care.
I learned that once my parents needed help with personal care and living arrangements it's good to try to follow their wishes but when it's not possible I had to have a tough skin and do what was best for them. Listen and sympathize. Develop a mantra that says, I know you're not liking this but this is the way it has to be. Don't argue or try to convince them that it's right. Be sympathetic. Let them vent. But stick with your decision. It's really difficult to be old and in need of care and equally difficult for family to deal with this. Be kind to yourself while knowing you have to make tough decisions.