Select Stunt Mode on the main menu, and choose any Jet Ski. Now, go to the Dolphin Park course. During the race, you must perform each of the following stunts while still passing through every ring: a handstand, riding backwards, standing and somersaulting back, barrel roll left off a ramp, barrel roll right off a ramp, flip off a ramp, and a submarine (diving underwater from a ramp). If done correctly, a dolphin will squeak when you finish.
Now, go back to the main menu and choose Championship mode with difficulty Normal, and enter the Warm Up. While choosing your Jet Ski, hold down on the control stick or pad, and press start. You'll now be riding on the back of a dolphin!