The GIF is helpful, but unfortunately it doesn't seem to be 60fps so I can't be 100% sure that I'm right about what I wrote below.
In most of these, you're parrying too early. Your shield is coming out before the punch hitbox comes out. The exceptions are the ones where you got hit (you were too late). The time that you got a powershield, you were very close (but still early). On the 4th Falcon Punch in that GIF, you did enter your jumpsquat, but I can't tell if you parried beforehand. If you did, you were very early.
It seems like you're trying to parry, then check to see if you were successful, then jump. Or you're worrying too much about making sure that your jump comes after you block the attack.
- Think of parry > jump as one action; there isn't enough time for you to confirm that you got the parry, since you only have at most 5 frames before the window to jump is gone.
- Make sure that you're not letting go of L or R before jumping.
- Try to time the parry so that you're hitting L or R right when the punch hitbox comes out, not before it comes out.
- If you're not trigger tricking your controller, you should. Hold L or R (whichever you use to parry) fully as you're plugging in your controller, or do it while resetting your controller with X+Y+Start. You won't be able to light shield with that button, but it will make parrying (and specifically jumping after a parry) easier.
- If you're using the 20XX hack pack loaded with Nintendon't, trigger tricking might not work. I have heard of a setting with Nintendon't that will allow it to work like normal, but I don't know how to do it. Dios Mios works fine.
- Remember that parrying is very difficult. Once you get it, it will be as natural as the rest of your tech skill.
Don't give up!