The falling piranha shot is a very common run killer.
The best strat is to start the area by running right and when you've cleared the platform you tap left to face left. If you do it right you should be close to the wall and the piranha should face right (most important part).
Now hug the wall by holding right and shoot a neutral egg in the while touching the wall above the piranha. The frame window for this is 13 frames.
The images below show the timings of first frame to shoot and last frame to shoot. It's also possible to stand next to the piranha if you release right just as you clear the rock, you can stand there for a couple seconds until the piranha face left again.
Most of this level is pretty straightforward once you have a good route down, it is just very punishing if you mess up a jump. Luckily, some quick fluttering can allow Yoshi to recover in most of those situations.
Skipping the goonie jumps shown in the video barely reduces the speed taken to complete the level, but makes the middle section of the first room much safer.
The area after the save ring is extremely deadly because the falling stone platforms tend to eat your jumps.
Finally, in the last area, staying one set of stone platforms higher than shown in the video is much safer, as a missed jump will no longer result in a death.