I also work with toddlers and have seen this before. This is very typical for this age group, as you know. It's so nice to hear that you want to help her bond with you by holding her. I can't stand it when I hear teachers say that they don't want to hold children because they are afraid they will get too used to being held. Children adjust so much faster when they can bond with you, then they feel safe to go off on their own and explore--as opposed to being ignored so that they don't get too attached.
One 18 month old in my class went to Chicago and stayed with her grandparents for a week while her parents went on vacation. The first week she came back, she was fine. However, the next few weeks after that, she behaved the same way as the little girl you are describing. What helped the most was that we asked her mom to bring in a few pictures of both of her parents. We laminated them and let her hold them throughout the day gave them to her when she was upset. We talked about mom and dad a lot and reminded her that they would come back.
Something else that helps a lot is having the parents create a routine when they get to your house. Have them continue to hold her while they put her things away, then hugs and kisses (sometimes a little phrase or something they repeat each day can help create some predictability or routine), say good-bye, then leave. Please explain to them that they are welcome anytime, but the longer they linger in the morning, the harder it makes it for the little girl. Lingering would show her that they are uneasy about leaving her and she will pick up on their anxiety. Make sure they stay positive so she will be confident that they are leaving her in a good place.
Please make sure they do NOT sneak off while she is not looking. This would make her anxious that they could leave at anytime when she isn't looking and make the problem worse.
I also had a little boy who did not want to be cuddled or held. He took a very long time to adjust. In order for him to know that I liked having him there and that he was safe, I would often smile at him from a distance. He eventually began to sheepishly smile back. This helped us create a bond and he eventually became more willing to let me hold him or sit on my lap.
Many kids like fun music, bubbles, engaging activities. Even something that she doesn't have to participate in but just watch, like bubbles, could help her warm up over time.
Remember that it can take 6-8 weeks for a child to fully adjust. I would let her parents know this so they don't expect her to adjust too quickly. Hope this helps!