I am the only one without obligations, but I have lived independently for many years with cats, but no husband. My brothers want me to move back in with my parents to assist them, but Mom says no cats,which pains me greatly as they are my family too, not to mention the abrupt loss of freedom and feeling as if I were instantly back in high school. It is a small town with little job opportunities so they are planning on giving me an allowance! Wouldn't it be easier on all of us if I transitioned into this role less abruptly by getting an apt. nearby and spending the time needed, and then later moving in or convincing them to go to assisted living?
Start reading here about children who move in with parents. Get your parents to consider assisted living. A parent with dementia will need more and more help, beyond one person's capabilities. And when there's two failing seniors involved and brothers who expect you to do all of the work, it's a recipe for disaster.
Work together with your brothers (and their wives?) for getting some outside help in for your dad and mom. Don't give up your life (or your cats)! You deserve independence and happiness too. I'm sure others will weigh in with some good ideas, but DON'T DO IT! There are other options that don't involve you moving in and giving up your independence and your precious pets.