Invincibility:
Pause the game and press Triangle, Triangle, Right, Left, Right, Left, Down, Left. If you did it right you'll hear a sound. Enter the code again to resume mortality.

Level Select:
At the title screen, press Square 5 times, Triangle 7 times, and Circle 3 times.

Move Options:
If you're maxed out on Options (four rabbits) you can change their alignment by getting enough capsules to choose Options again.

Power-Ups:
Pause the game and enter Up, Up, Down, Down, Down, L, R, L, R, Right, Left. You should hear a noise and when you unpause you'll have maximum power (not counting your shields). In a two-player game this code will power-up your partner as well. If you use this code with the original Parodius as well and you get a shield when you do it.

Save Your Game:
Let the timer run out and go back to the title screen. Choose "Save" at the bottom, press Start, then confirm the save. This will save your button configuration as well as your progress. Do the same thing with "Load" to load an old game