Eoinsmom, you hit the nail on the head! When I have a new child start I have a talk with the parents about encouraging their child to play on their own sometimes and tell them that it's important for children to learn to play by themselves. So many parents think they have to direct every movement for their children and I can spot it now very quickly in a new daycare child.
I also ask parents not to pick up their child at the first whimper, but to give them time to learn to self-soothe and get over whatever is bothering them on their own a bit to learn to control their frustration. It's hard for a lot of parents to realize that letting their children be an individual instead of an extension of themselves means letting go a bit even when they are babies. Some parents can't do it!
Talk to the parents.