How many times has she asked for a Saturday off at short notice?
I think it’s ok to ask for reasonable notice such as 2 weeks or so and if it’s not convenient, you need to say so. Working Saturdays is pretty standard for salon work so it won’t be any different elsewhere.
Unfortunately, most school mums don’t hand out party invitations until quite late but can she not ask a family member to help out or a friend who’s child is also attending the party to drop & pick up and return the favour another time? Usually, parents aren’t expected to hang around at a party and they’re normally 2 hours max at primary school age and held in the afternoon, so she could at least work all morning.