I know this is asking a lot from an employer... but do they give you time off to take care of the family member?
Most employers over a certain size have to abide by the Family and Medical Leave Act, which means allowing you time off to care for family members or your own serious medical condition. I often see FMLA paperwork in my job, and sometimes I glean important information from it. I've never had a reason to use it, but I know it is there if I need it. You don't get paid using FMLA, but your job is protected. You do have to pay insurance premiums, I believe.