Any suggestions in dealing with one parent that thinks they are capable of caring for the other parent when they both are clearly unsafe?

I went through this. Dad was healthy and mobile but had dementia and little short term memory. Mom was still somewhat with it but was almost immobile. They would agree to nothing. Not moving or any help whatsoever. It was a mess. It finally took a bad fall, hospital stay then I moved mom into assisted living. Got dad in a few days later.

You need to add more info about your situation so people can better advise.

In my opinion, these are signs that a spouse may neglect the other. Get in touch with your state department of health. There are services out there that can assist you. They may qualify for home health care, or at the least, some type of assessment from a county social worker.