If you have one mouse you have mice, they do not live alone,
with regards to them living under fridges, you need to act, one of the main problems, that comes with mice, is their love of chewing wiring, which often leads to fires,
a mouse can squeeze through any hole you can poke a normal pencil in, so quite small,
there is no point in trying to eliminate them while there is any food supply, all food must be stored in containers, glass is the only rodent proof container so buy lots of kilner jars,
clean all the cupboards out, anything not in a container, chuck out, disinfect all the tins, containers, take the kick plates off the bottom of the fitted cupboards, find all the items that have fallen out of over filled draws, cupboards, brush out, you will be able to see from the dropping how bad your problem is,
go outside if you have air bricks(you do, unless you live in a cob house) get fine mesh covers for them, fit properly,
anywhere pipes go in or out seal the gaps with proper pointing,
If you grow foliage up your house, make sure it does not go up as far as the sofits, as rats and mice use plants as ladders enter at the top, and come down in the walls to food source.
make sure the kitchen and anywhere else people or pets eat is vacuumed last thing at night,take the rubbish out at night, don't leave pet food down, make sure pet food is stored in metal containers,
If you feed the birds, stop or at least have something to catch any dropped grain from feeders.
get killing sprung traps, set along edges of room, keep pets out or set in cardboard boxes, with a hole at each end for rodent entry,
bate with crunchy peanut butter, by far the best bate.
Oh and do it as a present to your folks as they don't seem to want to tackle it, so just do it as a thanks to your hosts.
and keep ALL food in containers, and make sure the floor is vacuumed.
good luck.