Your parent's death affects what companion you get in the Presidio, around lvl 7 or 8. I think the companion there starts as lvl 9 and you acquire them after you get back to Gullet, if I remember correctly.
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During your adventures, you're gonna be robbing a little Monquistian fortress called "The Presidio".
When you are there, you'll come across a prisoner who knew your parents. You will help to free them and they will join your crew. Which prisoner you find depends on how your parents met their end.
Death by squid = Birgus Latro, a Crab Thug (Buccaneer) Death by storm = Milo Greytail, a Wharf Rat (Swashbuckler) Death by Armada = Gasparde de Vole, a Guinea Pig Guardsman (Buccaneer) Death searching for treasure = Lucky Jack Russell, a Dog Pirate (Privateer) Death by mutiny = Dead Mike, an Undead Pirate (Buccaneer)