No, parrying doesn't affect the damage of things like the Leo ring. In DkS2, many enemies have large recovery times after they attack. Hitting an enemy during these recovery windows gives you the bonus. As to how to exploit this damage? Use a shield with good stability and 90%+ PHYS reduction. Wait for them to use their stamina in attacks, then punish them for being dumb enough to challenge you. I do often use the Leo Ring + Grand Lance +10 to do well over 800 per hit.

But the Leo Ring isn't always needed. I'm on a quality build (more strength than DEX) and so weapons like the Pursuer's Greatsword or the "regular" Greatsword are amazing for counter damage. The way to use these non-thrust wrappings effectively to get counter damage is to outspace your opponents. Wait for them to attack and then slide into their DMs with punishing counter hits. Unlike DS3, there isn't an audio cue as to when you've counter hit, but you'll notice much higher than average damage.

This technique is most effective on bosses with slow recovery times (all except a small few in this game), so getting proficient at outspacing or shield usage is the key to success.

In general, the best weapons for counter hits are: Craftsman's Hammer (strike, and fast attack speeds, best suited for quality or strength builds), Grand Lance (thrust, deals nearly 190 counter damage worth Leo Ring), Chaos Blade or Blacksteel Katana or your katana of choice (DEX focused, R2 attacks trigger Leo Ring bonus), Greatsword or Pursuer's Greatsword (for Strength builds, best used after practicing outspacing techniques), Dark Crypt Blacksword (effective against later game mobs and bosses, gets an S scaling with Dark infusion, great counter hit damage, and is a strength build weapon), same with Smelter swords and Drakewing UGS, and finally a Mail Breaker makes a great offhand choice if you have a slow right-hand weapon.

Obviously anything that has thrust attacks like spears still work well with Leo Ring, and most spears have good counter damage modifiers. These are just a few suggestions that I've learned from playing around with tons of weapons. My personal favorite with Leo Ring is the Grand Lance, and the UGS that I listed (namely Crypt, Pursuer's, and Greatsword) are my favorites without Leo Ring. Hope this helps some!

And sorry for any misspellings. My phone's autocorrect is as autistic as the R1 spam Dark Sword users in DkS3. Again, if you have any questions, you can hit me up on PSN: The-Fenrir-LL. Cheers.

Learn to read

if you interrupt an attack/"maybe roll i dont really know" of your oppent with an R1/R2 it deals counterdamage

Yes

Uh... no. All damage types.