If two bedrooms share a bathroom, is it allowed that persons sharing bathroom be of different sex?

Hmm. That's interesting. Never heard of that. I would feel a bit awkward with that set up if that were my mother. Not that she's in any sort of immediate danger - nothing like that - but it seems strange. I guess if there are locks like freqflyer said, then there shouldn't be any sort of trouble. I would also be more aware of cleanliness/hygiene sort of stuff. Make sure they clean it well.

Hopefully there are locks on the bathroom door for privacy.

Only by mutual consent.