Hold Z + R + L when the level description appears at the start of a level. Note: The invincibility only lasts for the current level.
Go to the ant hill from the main screen. Then, hold C-Up and press R at the "Flick's Journey" screen. An arrow pointing to the right will appear at the bottom of the screen to confirm correct code entry. Press Analog-stick Right to scroll the screen in that direction to access any level. Alternately, hold C-Up + C-Down + C-Left + C-Right + Z and press R.
If you cannot find an invincibility plant, make a helicopter plant. Jump on it, then jump off. Flik will still be spinning and is invincible. When you run into enemies, they will die. It wears off if you jump or after a short period of time.