I think it's an important thing to think about, depending on your family circumstances, at least if you live in the US (I don't know what options/resources are available other places so cannot comment on those). My dad is declining with dementia in a nursing home, which is paid for by Medicaid (medical/LTC for the very poor, not to be confused with Medicare which covers medical care but not generally LTC for those 65+). Even if I wanted to and chose to SAH, i.e., quit my job, I couldn't care for him at this point, but I wouldn't choose to, either, a decision that reflects his earlier behaviors, i.e., he's not a very nice person (he's also not an ogre, just to be clear -- and I do tend to him, and his care, in the nursing home, i.e., advocate for him, visit, etc.). OTOH I'd drop everything, including my financial well being, to care for my mom, who IS a wonderful person (but who'd be horrified if I did that), but who's also a planner, i.e., has savings and, among other things, LTCI (my parents are divorced, so their fates are largely separate). But that said, watching my dad's decline and those of others of my parents' generation has shown me just how much care can be required, and one motivation for achieving FI is being able to leave the workforce if my mom needs care. And while my wonderful parent happens also to be a planner, and the not-wonderful parent is/was not, I don't think those characteristics always line up, Mustachian ideology notwithstanding, so it's not hard to imagine having a parent who deserves personal care but hasn't planned ahead to allow for it.
I've seen 2 friends navigating this general issue with unexpected challenges that affect their budgets. One whose dad, also a dementia sufferer, was living with her felt she had to find another place for him after he assaulted her -- she lives alone and didn't feel safe. He doesn't require skilled nursing care for now, but cannot be left alone or manage things like preparing meals independently, and even the place she has found has cost more $$ per month than her dad has in income, so she's making up the difference which I think runs her maybe $8K per year (and when he requires more care, who knows?)? Another's dad is dying of cancer and my friend is using FMLA to be able to be with her -- of course, not knowing how long her dad will be here, she wants to spend time with him, but she doesn't know if the FMLA will be enough, or not, and if not, whether she can afford to quit her job.
As for what the OP should do, I'd say it depends pretty significantly on (a) what your other responsibilities are e.g. to kids if you have them; (b) your parents and your relationship with them; and (c) what arrangements they've made, or could make, to ensure that they have the care they want access to if they need it.