Unlock Blue Hat
Beat the game once, and Parappa wears a blue hat

Unlock Dog House
Beat each level while wearing the yellow hat and Parappa gets a new dog house. Go there immediately to listen to some tunes. In the dog house, you can listen to tracks from any level on which you've earned a "cool" rating.

Unlock Pink Hat
Beat each level while wearing the blue hat to give Parappa a new pink hat.

Unlock Yellow Hat
Wear the pink hat and beat each level. Powered by Coranto