You can jump in the water anywhere by the dock when the parrot says to.
To get the map piece :
First get the cracker from the upper left inside Petey's Pub. When you see the parrot at the Trading Post, it wants you to jump into the sea. When you do so and return, it is at an upper window. Now it wants you to "clog the vent", to stand on top of the smoking chimney of the Pub. It is then on top of the palm trees at far right. It wants you to talk to the pirate atop the tower near the docks. You will find that it is his parrot. Use the cracker there, and the parrot will fly over to him. Click the pirate to get the map piece.
The only time you dive in on purpose is to follow the instructions of the parrot at Parrot Port. When you jump into the water and return, the parrot has moved to a higher window. (see related question)
parrot port.
If you look at the island in the middle it is to the right of it
Parrot Port
you dive to were your ship(raft)is and you swim till you get to sand
Usually Parrot Port, for between 32 and 40 doubloons.
It is the island directly to the right of Fort Ridley, in the 3 o'clock position on the island map.
It's on top of the pub (on the chimney).
You find the navigator in the second floor of the bar at parrot port.
Parrot Port has the highest price on medicine. It is the island in the easternmost island.
Parrot Port- it is the island to the immediate right of Fort Ridley
At Parrot Port, the "vent" is the chimney on the roof of Petey's Pirate Pub.