Bonus Parappa Levels
Beat the game, then return to the level selection screen. Press Right until the bonus Parappa levels appear.

Parappa and Lammy Two Player Levels
Beat all of the bonus Parappa levels.

Make Your Own Lyrics
At the main title screen when you see Lammy with her guitar, reset the PlayStation immediately. When you see the title screen again there will be another option. If you select it you will be able to create your own music.
Submitted by Pickle. 

Go to: Um Jammer Lammy Info Page | nGaming Home Page.