Unfortunately potty training is very much a personal (perhaps controversial) thing. Everyone has their own system and everyone has their personal beliefs as to how it is best done.
What you need to do is determine what YOU want and what YOU are comfortable with. Then let the family know (this should be in your contract). What the family sees at home isn't what you see at daycare. The child may be ready at home and the parents are more able to be leading this process with just one child to train. But at the end of the day it is your house getting peed on and your day getting interrupted.
What have you done with other children in your care? What is different with this child. Are there ways to change that to make it easier on you but still accommodate the way the family is trying to do this?
Maybe set a timer to go off every hour and when the timer goes off the child has to go to the potty on his own to pee. This will free you up and will still remind the child before he pees on your carpet.
But, ultimately your contract should clearly state what you allow and the family needs to default to that if you are not able to accommodate the child otherwise.