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.