https://gamefaqs.gamespot.com/snes/588362-gradius-iii/cheats

the relevant portion:

<B>Stage 3</B> - When you reach the lower level of the stage, there is a platform in the ceiling that drops down. When it drops down, fly just below the part where the platform was.
Prerequisite: Destroy all of the ground targets in the stage up to that point.

<B>Stage 4</B> - In the last set of Moai heads (they look like Easter Island heads), there is one laying down which will lift itself up. Fly behind it.
Prerequisite: Nothing must be highlighted on your powerup bar.

For the "destroy all ground targets," you need to kill all the dispenser enemies.