I have three data points. Why does Prism tell me I have too few points to fit a second-order polynomial curve?

Prism requires at least one more point than the number of parameters you are fitting. So if you are fitting three parameters it wants four or more points. But three points really do define a model with three parameters in most cases. To work around the limitation, copy and paste so as to duplicate your data--instead of three points, you'll have six with each point entered twice. Now you'll be able to do what you want.