Play as Sunny or the Ninja:
Complete level 1 with a least 3,000 and a cool rating.
Complete level 2 on Good with Cool Status flashing.
Complete levels 3 and 4 with over 2,000 points each and a good rating.
Complete level 5 with at least 4,000 and a cool rating.
Complete level 6 with at least 2,000 and a cool rating.
Start a new game and you'll get the choice.

Bonus level:
Complete all the levels with the rating of 'cool' to unlock a bonus level.

Change Parappa's voice:
Complete the first two levels with the 'cool' rating.
Then press this combo:
X, X, X, T, C, T, S, S, Right, Left, S, X, and finish the next level as usual.
On level four press:
X, S, T, C, X to change his voice.
To cycle through the voices hold R1 and press C or T.

Bonus background image:
Hold 'Up' and keep rapping on level 1.

View movie sequences:
Complete levels 1-5 with a 'cool' rating.
Level 6 with an 'Awesome'