Ah! I think I’m beginning to see the pattern of your matches: Parry into a rune that does great stagger (like fire flurry, with 20 bonus poise damage and DoT), then either a backstab or a chain stagger.
Chain stagger is hard to counter. There are some protections against chained stagger build-up while you are already staggered. In PvE, staggered targets take 0 poise damage.
This protection works most of the time in PvP, so your friend would have to roll-catch or guess your wakeup. You can prevent the roll-catch with Blink, by mixing up dodge directions, by using quickstep to begin with, or with runes with long initial i-frames (like Spirit Edge’s Evasive Strike); or mix up with a point-blank knockdown rune or parry.
Additionally, any damage taken while staggered will unstagger you (and won’t inflict hitstun itself). It can be hard to capitalise on this in the middle of a flurry rune, but if you are fast enough this can allow you to escape. Not sure if you can buffer an i-frame rune or dodge, but you might be able to mash to execute it between first and second hit.
As you say, preventing that initial parry is probably safer. I’d try spacing and a rune with guaranteed knockdown (Crush, Piercing Spin, Scream) to bait and punish a failed parry. From neutral, many pokes are fast enough to be unparriable on reaction (even on heavy weapons, e. g. Summer’s Sting running attack), so your friend would have to predict you.
Light weapon flurries bounce on heavy shields, so if you can tank the first 30-ish poise (tough one, you’d have to play keepaway once you have any build-up), you can convert the bounce into a knockdown with Piercing Spin or Scream.
If he uses health to fuel his runes you can maybe out-trade him with a heavy weapon (granite in a fast 2H hammer or axe). If you can stack attack speed and land a charged Pig Sticker attack or an unparriable bleed rune without being parried, bleed stops the aggression of low-health builds quickly.
Parry fishing is strong currently, and unless you know your safe pokes the less aggressive player wins.