I was thinking if there were others in the same area that were taking care of their parent or parents it would be interesting to match similar situations and perhaps the caregiver could have their parent stay with another caregiver while they shop and they could trade off. I know it is difficult to get the parent out of the home environment, but it might be worth a try. It could be beneficial in providing some friendship for the parent that they might otherwise not have. This might not work well for severe dementia patients, but for just elderly with slight dementia it might work.
I think you do have to think about the legal and financial ramifications should anyone become injured or die in your home. Is a good idea but the caretakers may need to stay with them, rather than leave.