These are pulled (and slightly edited) from the gamefaqs page contributed there by user gs68: https://gamefaqs.gamespot.com/snes/588362-gradius-iii/cheats

Stage 2- When you see a hole in the wall at the bottom of the screen, fly down there. Prerequisite: None This stage does not seem to have any use in speedruns since it comes right before the boss.

Stage 3- When you reach the lower level of the stage directly after the active volcanoes, 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 dispenser enemies (the ones that spawn other ships) in the stage up to that point. This bonus stage saves a massive amount of time over finishing the stage normally, don't miss it.

Stage 4- In the last set of Moai heads , there is one laying down faced up which will lift itself up. Fly behind it. Prerequisite: Nothing must be highlighted on your powerup bar. I found this one to save about 3 seconds in my regular Easy runs, worth it if you can get an early entry, though I haven't entirely tested how much difference the time of entry affects the time saved.

Stage 5- Fly just below the ceiling before the narrow corridor. Prerequisite: The hundreds digit of your score reads 3, 5, or 7. This is useful for speed only runs, though is still slower if you can kill the boss fast.

Stage 7- Just after the long downward slope in the second half of the level, stay close to the ground and fly into the indent in the wall that comes up. Prerequisite: Fly through the smallest gate. This one is useless as it is so close to the end of the stage.