I agree that your friend shouldn't have to pay the month's notice as the caregiver should have given a month's notice that a change like a pet was going to happen, in order to give her clients a chance to give notice if they didn't like the change - the same procedure that the caregiver should have followed if she was changing other major things - fees, hours, etc.

Does it say in the contract that the daycare is "pet-free" or just the ad? If the actual contract says "pet-free" then the caregiver is breaking her contract. If just her ad says that, then your friend will have a harder time getting out of paying the notice period. Also, does the provider know that the child has allergies? Was this discussed during the interview? As in, was the condition that the daycare be and remain pet-free discussed and agreed upon prior to the child starting? These could help sway your friend's case if she does choose to fight the notice.