What is Charlotte parodying when she tells the story of her cousin catching the fish in her web in Chapter 13?
To me, there are a couple of things that you could say she is parodying.
First, the whole thing is a parody of a "fish story." Fishermen are notorious for telling tall tales about their exploits and this sounds very much like one of those stories.
Second, most of the time that Charlotte is actually narrating the action, she is parodying the blow-by-blow report of a boxing match. "First she threw a left around the tail. The fish lashed back..." That whole paragraph sounds very much like someone announcing a boxing match.
So you have two levels of parody here that I can see. A more subtle one (the fish story) and a more obvious one.