I am a caregiver like you. But I think the answer is that parents are legally responsible for their children, but adult children are not legally responsible for their parents. Funny you should ask this question. First of all you are not alone in this dilemma. It comes up all of the time.
Experts recommend holding a family meeting, early on in the caregiving situation if at all possible.
But really any time is fine and they ought to be repeated on a regular basis to share thoughts, grievances, etc. It is often wise to have a 3rd party present, such as a geriatric care manager, a clergy person, etc. Regarding disagreements a professional person serving as facilitator can help you come to consensus. On the other hand distance can sometimes provide a fresh perspective. The trick of course is in how we communicate our thoughts and feelings.