My issue with this is that she lied. If you were babysitting, especially a family member, and the child lied to you intentionally, would you tell the parents? That's really all I want to know, please.
Yes. Yes I would have told the parents the situation. Because I would expect that, were the situation reversed, my child's caregiver would report any breach of our rules (such as no lying) to me. HOWEVER, as the parent I would also want to know both sides of the story before the word "lie" was ever used. Sometimes what we see as a lie, a child sees as a tough decision. As the caregiver, I would have told the parents in a simple way what happened, and let the parent make the judgment call.
"I thought you should know what happened with your DD. She did not care for one of the foods I served her at dinner. Even though I knew she did not care for the sweet potatoes, I asked her to eat three bites and told her that if she chose not to eat three bites, she would not get dessert. Later I found her sweet potato in the trash wrapped in a napkin."
See how that is non-confrontational? It doesn't put any blame on the child, the parent, or the caregiver (yourself). Simply present the facts. Let the parent do the parenting.
Maybe in the child's home environment, politeness is stressed more so than food choices. The child then would not have wanted to be impolite and refuse food, so (being a child and lacking the ability to make complex, mature decisions) she would then decide to do the polite thing and make the sweet potatoes disappear. However, she would have known food choices are not a big deal so, while trying to be polite and at the same time make the food disappear, she did what seemed logical and hid it in the trash. Admitting she threw food away would be impolite, but so would lying, so when you asked her about it she was put on the spot and said she ate it.
Maybe in the child's home, honesty is more important than food choices. In that case, the child's trashing of the potatoes would not be a big deal but her ensuing lie would have repurcussions.
Whatever the situation, it is not your job to make a judgment call on someone else's child's behavior. As a parent, I would want to know what happened -- the whole story. I am willing to bet that the child's side of the story is quite different. You put that little girl on the spot. She did what she thought would make things ok and now you're going to rat her out as a liar to her mother.
Presenting your side of the events in a calm, unemotional, nonjudgmental way will help the child's mother decide on the appropriate course of action. Then you need to step back and look at your own behavior. If you hadn't pressured the poor kid to eat something that she expressed she didn't like, especially after she did eat the protein (chicken), and food issues are not pressed in her home, she would not have felt the need to hide the food. I believe she was trying to either please you, or shut you up. Either way, it failed miserably and she most likely feels pretty bad about it already. Then you put her on the spot by asking her if she ate it, when you knew she didn't. Most 9-yo's won't say, "Nope, I threw it away to get you off my back." By that point, she had already trashed the food and thought it was over, she didn't know how to proceed when further questioned.
At this time I'm not sure telling her parents would make much difference. The mother has undoubtedly already heard about it from her daughter and guess what? You're the bad guy in that story, not her. Any attempt by you to tell your side now will look like you are either calling the kid names or trying to defend yourself against a little girl. If it was right away, telling the parent would have been fine. Now? Just let it go.
It is ironic that the dessert in question was healthier than the refused food.
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to him. I am not insulted nor do I make a point of cooking it just top prove a point. Should I treat the children at my table any differently becasue they are children? And I can? 
