The biggest virtue of this book is that the author sees a major problem with the standard interpretation of Parmenides and his influence. Its biggest defect is that she ignores all the OTHER major problems with that interpretation.
The standard interpretation says that Parmenides was brilliant, he believed that only one thing existed, he had an enormous influence on his successors, and his philosophy received its first genuine refutation in Plato's Sophist. The problem that Professor Curd finds is that none of his successors ever produced any arguments against the second claim. They simply assumed that there was more than one thing, even though they seemed to accept other things that Parmenides had argued for. She then concludes that Parmenides did NOT believe that only one thing existed; instead, he believed that whatever exists can have only one nature. Yet, is it really likely that this is all that Parmenides believed? The author mentions the passage at Parmenides 128c-d where Zeno talked about how Parmenides had been ridiculed by others, but seems unconcerned by it, yet is it really likely that believing that everything has one nature would have elicited ridicule? Moreover, she seems unaware of the Commentary by Proclus, where he tells what these people said: "if being is one, then Parmenides and Zeno do not both exist at the same time" (619). Naturally, we don't know if Proclus was right about this, but Prof. Curd seems totally unaware of it. The reasonable conclusion is that Parmenides believed that only one thing existed. By going against this, Prof. Curd has taken us two steps backwards, even though she has taken us a step forward by pointing out a major flaw in the standard interpretation.
The big question for those who accept either the standard interpretation or Prof. Curd's revision of it is: Did Parmenides have an enormous influence on his successors? To see that he did not, it is sufficient to mention just a single name: Cratylus. Cratylus was a Heraclitean and paid no attention whatsoever to the strictures developed by Parmenides. People who accept the standard interpretation almost never say anything about Cratylus, for obvious reasons. Prof. Curd herself mentions him just once, in a footnote. But not talking about Cratylus will hardly make him go away.
In addition, here are just a few of the facts about the standard interpretation that I find troubling: Protagoras is said to have written a book refuting Parmenides, Plato hardly mentioned Parmenides in his early and middle periods, Aristotle did not record that Plato was influenced by Parmenides, Plato seems to have a refutation of Parmenides at Euthydemus 286a-d (which is long before the refutation in the Sophist), and finally, the movement begun by Parmenides DIED with Melissus. Why? Why would a viable and powerful philosophical movement just die?
I invite Prof. Curd or anyone else to give a convincing explanation of these points.
