How would this story be different if it were told only in chronological order as it is in Part III? What do Parts I and II contribute to the details and information provided in Part III?

Parts I and II are not told chronologically, and they therefore force the reader to engage more with the text in order to understand who the characters are and how they are related. Part I actually takes place in the middle of the story, at a concert right before the affair starts up again. Part I introduces all of the characters and gives an idea of their relationships to each other. But it does not give much backstory.
Part II gives more history and context, specifically around the events that took place once the affair had ended for the first time. This section gives more insight into Anna's mental state, something that the reader can better understand and sympathize with because of the way in which the story was told.
Part III serves to tell the whole story chronologically, now that the reader is more familiar with both the characters and the series of events. On its own, this section might not have caused readers to feel any sympathy for Anna, but because the first two sections jump around in the story, this section feels more like it is summing everything up, and the reader is able to feel sympathy for Anna.