If you feel like you can bring the subject up, I think it would be worth it to give it a shot talking to mom or dad. Just mention that he doesn't seem very happy a lot of the time (as hard as it may be, try to avoid the word 'whiny') and the only thing that seems to soothe him is being picked up. Explain that it is simply not feasible in a multi-child care environment, and ask what other methods they use at home. If you get the 'uh, uh, well....' then you know that it's for exactly that reason! If you can, ask that mom and dad work to find other ways to make him happy at home, like using activities to distract him, not just cuddles.
The bottom line is; for me, if it affects our day-to-day then it is a possible termination scenario. Like the others said, it's not fair to you or to everyone else in your care to deal with a whiny, boo-hooing little person for 8+ hours. Ugh... I have a 2 year old that can get insanely whiny and sucky if she is overtired, but she is old enough that I can say to her "When you are done crying, you come and see us again." I put her on my daughter's bed (door is open, light is on) and she just comes out when she is calm again. But I would not do this with a 15 month old, he is a little too young.