Yes, Cleaner Wrasses may pick up parasites on fairy wrasses but they do not accept foods offered and do notlive long in the aquarium.
Here is a new comer, but a popular species, C. lineatus, 10 cm long. It came from Australia and soon displayed in the tank without any female.
This is close to C. laboutei from Australia, New Caledonia, Loyalty Islands and Vanuatu.