When do "multi parries" happen?

As of Update 3.0.0...

Both parries on projectiles and multi-parries got an overhaul. From the patch notes: "Shortened the downtime when performing a perfect shield against projectiles." and "Perfect shields will no longer activate multiple times when the shield button is released once. This was creating more downtime than a single perfect shield."

So the values in this answer and information on multi-parrying in general may no longer be correct. I'll leave the below answer for historic value, but until things are tested, take it with a grain of salt.

TL;DR

Multi-parries happen either because you've auto-parried (a move has multiple hits that come out within 1-2 frames of each other) or because after the first parry, you timed another parry for the next hit using the 3 frame advantage from the first parry.

Before we get started...

For this one we'll have to dive into some frame data. As a quick recap: Smash Ultimate runs at 60 frames/second. That means a single frame is representative of 1/60th of a second (if there are no frame drops/you're not playing online with terrible internet).

All information below is pulled from this awesome write-up on Twitter by @Chiroz (they have videos to go with it too, so if this doesn't make sense be sure to check out their thread!)

With that out of the way, let's get to it!

A look at the frame data

It takes 3 frames to begin shielding. Once shielding, you can drop your shield at any time. After dropping shield, it will take you 11 frames to act. We call these "shield drop frames." Shield drop frames can be cancelled with a jump at any time.

If you're hit with a move during the first 5 frames of shield drop, it'll result in a parry. If the attack wasn't a projectile, parrying freezes the attacker for an extra 11 frames and freezes the defender for 8 frames (and cancels the shield drop frames). This results in the defender acting 3 frames faster than the opponent.

The 8 frames of freeze is likely for cinematic effect and to give the defender an 8/60th's of a second to react to the fact that they parried.

Fully parrying multi-hits

This is all well and good for a single strong attack, but things get a little more complicated when you're dealing with multi-hits.

It seems that if you parry the first hit of a multi-hit attack and then next one comes out in 1-2 frames (12-13 frames if you take into account attacker freeze), you'll auto-parry it (this is the case with Pit forward-air/Yoshi down-air/Ivysaur neutral-air/etc...). If the hit comes out later however, you can shield and parry again yourself. Again, you have 3 frames from the moment you can act before the opponent will continue their multi-hit and it takes 3 frames to begin shielding and then drop shield for a parry. So depending on the multi-hit, you'll need to time the second shield for a second parry.

To practice this, I'd suggest going into training mode against a Zero Suit Samus and setting the CPU to repeatedly perform forward-smash. See if you can time a parry on the first and second hit!

Partially parrying multi-hits

So parrying a full multi-hit is just a matter of landing the first parry and then getting the timing right for the rest of the move (or auto-parrying). But what if you take the first hit of the multi-hit on your shield?

It's still possible to drop shield in the middle of the move and parry the rest of it, but the rules are a little different. When a move hits your shield, you'll have Shield Stun, which will increase the amount of time before you can shield drop. However, if you shield drop some frames after coming out of shield stun, there will be some amount of frames that you won't be able to parry! The formula for this is still unknown, but it'll change from move to move.

This means that some multi-hit moves are not parry-able once you've taken a hit of them on your shield. Fox down-air is an example of this--you can drop shield midway through the move, but you'll get hit with the rest of it even though it's within the first 5 frames of shield drop. And other multi-hits can be furiously difficult to parry after taking a hit on shield. Captain Falcon neutral-air for example, ends up giving you only a 1 frame window to drop shield to parry it without being non-parry-able.

I haven't tested to see if Zero Suit Samus's forward-smash is parry-able if you shield the first hit, but I know that Chrom's neutral-air definitely is (and it's not too difficult). So if you have a friend, you can practice parrying only second hit of the move after taking the first on your shield.

Update

It looks like someone's actually put together a Google Doc of all multi-hit moves in the game and how to parry them/which will auto-parry. You can find it here.