He wants to put them both in a nursing home. Both parents have early stage dementia, but still function fairly well. Mom trusts this son and doesn't believe me. I've been their caregiver for over 2 years at a tremendous sacrifice financial loss to me.
You might suggest that both you and your brother meet with parents physicians jointly and discuss your observations, etc. with the doctor and get his professional input. Gladimhere also had a good suggestion on an independent third party evaluation.
Maybe your parents want to go to residential care and are open to the idea but aren't saying that to you expressly.
Are you and brother open to getting additional home care/assistance for parents so they can manage in their home a little longer?
This is hard. Please try to have a civil conversation with brother and ask him to visit some care facilities (memory care, AL, NH) with you (without your parents) so you can evaluate ALL your options. Then have a conversation with your parents, narrow down options to 2-3 places; take parents for a visit and enlist their thoughts/needs/choice in the process.
Personally, I think the care facilities are the best value over home care (especially if 24 hrs home care would be needed). Most offer a safe, clean, active environment for seniors with meals, transportation, outings, barber/beauty salons. Home care can keep the senior in their home for longer periods; but it is expensive option and there is still the need to ensure the home is safe/secure and accessible (trip/fall/slip prevention; shower accessibility, stairs, etc.).
Well, which is it? Is Dad functioning fairly well or dying?
You're trying to make money the issue, when it sounds more like jealousy. Somehow your 3 brothers all inherited money from their inlaws? So what? (Are you sure it wasn't their wives who inherited the money?)
If Mom could live many more years, can you afford, (since you've already had a severe financial loss) to keep her at home for all those years? Will you be able to work and maintain an income to provide for them? They would probably qualify for Medicaid. It isn't quitting or giving up to move them to a facility where they can receive many visits from their loving, well rested daughter who isn't trying to wipe their butts and make them take showers when they don't want to. If only 2 years into the process, you are already labeling yourself burned out, you may not last the whole of it anyway. It's a marathon, not a sprint.
By the way, word of loving advice here - you need a best friend who is closer to your age and who you can get out of the house and do things with and relate to. As does your mother if she's still able to maintain a friendship. Having Mother as your 'best friend' is not a healthy relationship.
She trusts her sons and doesn't believe you about something, but I'm not sure what.
Doubt that all the legal problems and certain-to-be rift between her children would make her happy.