Did I miss the screaming bit?
It's just common courtesy for the neighbour to ask if his wife's carers can park on his neighbours' driveway. Agree it's not the fault of the carer, although I would think they'd be a bit suspicious of the fact that they and a woman across the road did this, then had to move cars before the neighbour got home.
Maybe when he comes round you can ask him to let you think about it for a few days but meantime they need to use the street. But the neighbour with 3 cars is incredibly rude.