Oh there be many parallels, matey.
Many named references, character references, catch phrases, etc.

I was trying to think of the more subtle ones, such as the impersonation bit, how Jack acquired his bits of costume (the bandana and sash), his ability to negotiate his way out of any situation (rather than fight), the "P" brand, the pearl references leading to the name of his new ship.

And this was the interesting bit, I thought. In POTC I, Gibbs tells Will that not much was known of Jack Sparrow before he showed up in Tortuga with the Black Pearl. As the sequels progressed and it became clear that Jack's father was an infamous pirate, how could Jack have gone unnoticed?

I think Ann settled that here by having him "break" the code, be banished from Shipwreck Cove and become a merchant seaman. Perhaps, Teague was so ashamed of Jack's actions (setting the rogues free and joining them), that he forbade anyone to mention Jack's name. So for 5 or more years, he was forgotten until he showed up with a newly painted black vessel.

I thought that was a very nice touch.

My thoughts on this question are going in a different direction. With Ann filling in the backstory of Jack we see how the stories of POTC connect. More parallels are seen after we see the development of our hero becoming our Capt. Example - his wit, his loyality, and his shrewdness of getting out of jams. His appeal to the ladies seems to have been born in him.
Guess my thoughts have come full circle and I do see the parallels!

The parallel for me was in seeing how Ann remained true to the spirit of the Trilogy - particularly in the way she revealed certain things about Jack, but still left some key elements as a mystery. Like the films, questions are raised, possibilities suggested, but not every question is answered. I like that. Also, just like the films, the story contains elements of fantasy that intrude into the real world.