I think it is perfectly reasonable that a pet-owner does not want their pets to be outdoors, where they are at risk from coyotes or being hit by cars, potentially resulting in $$$$ vet bills and/or the death of a pet(s).

I think it is also perfectly reasonable to ask that an allergic person not be subjected to an allergy trigger, given how deeply unpleasant allergies can be, and that allergies often get cumulatively more severe with each subsequent exposure to an allergy trigger.

The solution would seem to be

"Mother and daughter stay
at a hotel
or motel
or a private room in a youth hostel,

sister meets them at
a cafe
a public park
a library
a playground
a museum
an art gallery
a beach
a public pool etc etc
to spend time with them sans pets."

I'm not sure Prudence is in the right with her "just take allergy medicine" because:

a) many allergists advise that allergies can get cumulatively worse with each exposure, so repeated exposures can result in a mild allergy becoming a life-threatening one;

b) allergy meds can be quite sedating;

c) long-term use of some types of allergy meds (the anticholinergic ones: not all allergy meds are anticholinergic) signficantly increases the risk of dementia[1].

[1] https://www.health.harvard.edu/blog/common-anticholinergic-drugs-like-benadryl-linked-increased-dementia-risk-201501287667