When a Slasher makes the third swing in their attack sequence, parrying it doesn't actually prevent the damage. When I successfully parry the swing, it says "Parry" on the screen, deals damage to the enemy, and triggers parry-based effects such as the bleed effect from Blood Shield, but I still take damage from the attack as if it wasn't parried or blocked. I've verified this repeatedly with lone Slashers, so there's no chance that the damage is coming from another nearby enemy.

I believe certain shields do successfully prevent the damage from the Slasher's third swing by interrupting the attack sequence entirely. This includes the Cudgel (stun) and any shield with the "freeze on parry" modifier.