I'll start with TMNT Mutants in Manhattan. Despite being a very average action game, I found it's parrying mechanics quite unique. It also has two types.'
The first I've praised before here. When you block the turtle withdraws into their shell. When you release the block button they pop out. If you time this with the connection of an enemy attack, they will parry it. It's really cool how they did this.
The only problem is, there's never really any need to use it. I only ever used it because I wanted to. Shame because it's such a unique way to parry and fits the theme of the game perfectly.
The second method I would hesitate to call a critical or a parry because it blurs the lines. Basically, when you dodge at the last second, the turtle will do this dash in an arc around to the enemy's back, then jump onto their shoulders and start pounding the enemy with his fists. You mash the attack button here.
While in this state, the enemy will flail around allowing your team mates to get in free attacks. It's pretty much like a stun, because they can't attack until the duration has ended.
So at startup it's like a Bayo dodge, but then it becomes this special attack. See what I mean? I'm not sure if it truly qualifies.
What do you think?
In any case, the dodge one is actually useful, because you'll use the dodge often, and the reward is great. The parry does open the enemy to a follow up of your choosing, but you don't have anything better than the dodge reward.
Found a video.