Unlock Blue Hat:

Beat the game once and Parappa will don a snazzy blue hat.

Unlock Dog House:

Beat each level while wearing the yellow hat and Parappa will get a new dog house. Go there immediately to listen to some tunes. In the dog house, you can listen to tracks from any level you've earned a "Cool" rating on.

Unlock Pink Hat:

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

Unlock Yellow Hat:

Beat each level while wearing the pink hat to give Parappa a new yellow hat.