John Hobbins at Ancient Hebrew Poetry points to parallel structure of Genesis 1 and Genesis 3. In both cases we have a temporal clause, a subordinate circumstantial clause, an action, and a resulting state. (See his post for more detail.) He then asks me the following question:

Hey, Kevin Wilson, online specialist in ‘P.’ What’s going on here? Is this evidence that the author of Gen 1:1-3 had Gen 2:4b-7 before him? Vice-versa? Or is it just a remarkable coincidence?

I would probably answer “none of the above.” If one is relying on the other, I would argue that J is relying on P, because I think P is earlier than J (see some of my arguments). But I don’t think we are forced to see such a relationship here. But I also don’t see the similarities as just coincidental.

It seems to me the structure can probably be explained simply as the way one starts a creation story (or perhaps any number of story types).  You state the time when the story took place, the conditions that obtained at that time, and what the protagonist did. The origin of this structure in Genesis 1 and Genesis 3 is probably more form critical than source critical.
Now, I have absolutely no parallels to back this up from other biblical stories or comparative data from Mesopotamia or Egypt.  Nor do I plan on hunting it down. This is nothing more than a hunch, but I doubt this structural parallelism is indicative of literary dependence.