After from the first game, Parappa lives in a place called Parappaville, which is (presumably) named after someone else. Considering that people who have cities named after them tend to be people of importance, perhaps part of the main character's obsession with proving himself is tied to sharing a name with this person.
The noodles Parappa won at the beginning of the second game are labeled "Beard Noodles." Which, given that Colonel Noodle is the Beard Burger Master's son...
Chop Chop Master Onion is a rap master with the head of an onion. He's a rap scallion.