Carpet or no carpet, running around naked with the potential pee or poo accident isn't sanitary or practical to expect of your daycare provider.
I agree with you....a child has to be able to communicate either with words or signs that they need to go, be able to undress themselves and be old enough to realize when they need to go.
Toilet training is never fun and it's even harder with part time children I have found. The key is to be consistent though so I would talk to the parents and explain why their approach is not practical for you and because of that, it will actually hinder their child by having two completely different techniques of toilet training. See if you guys can come up with a different approach that works for both parties and hopefully the training will come along quicker and smoother.
There is an article on pinterest about potty training your child in 3 days and it's approach is to let them go naked....I wonder if they read that! lol