I would ensure you check with other providers (as you are obviously trying to do). I would not take the words of one parent for what is the norm. Especially if this family is just starting to look into care for their first child. Many families enter their search thinking they pay the caregiver for the hours worked and if they keep the child home they don't have to pay etc. They would not be a good source of information for making major contract changes.
I personally think home caregivers deserve paid time off. While I think we also deserve paid sick days (especially when it is their children getting us sick) I prefer to have sick days unpaid as it makes me a bit less stressed with a last minute closure.
What ever you decide, never waiver in an interview. You work from home in order to work on your own terms. Your families NEED to agree to the contracted terms as set out. If you show any sign of being uncertain about them you lose credibility in being able to stand by them later on.