As everyone else has said, that PLD was a moron and extremely rude. Tanks that are rude for no reason are a thing, sadly (as are people being rude to tanks for no reason).
I usually tank (as PLD even) and I welcome DRG stuns, especially DRGs who are smart with their stuns. Enemies with no big stunnable attack - stun as part of your DPS rotation. It's part of being a DRG and there's nothing wrong with it. But, some enemies have a big attack that's stunnable (Manor Maidservant in Haukke NM, the mobs in Haukke HM that corrupt the woodwailers, the Charry Harriers in Snowcloak, the voidcalling cultists in AK, the bees in Qarn NM, the snakes in Qarn HM, most bosses, and so on. For those enemies, stunning them is still fine, as long as you stun their big attack. When I play DRG against those foes, I don't stun as part of my DPS rotation, but I do stun when those abilities come up.
The difference between you stunning as a DPS and the tank stunning as a PLD is that stunning increases your DPS while stunning decreases the tank's DPS. As a tradeoff, your stun is shorter than the tank's. If the content is hard for your group, the PLD's stun is better, but if it's easy content, your stun is better because it increases party DPS and shortens the fight a bit. So, the tank saying he has to stun and you can't is like the tank saying that he's terrible at tanking and needs the longer stun to survive. Just keep that in mind.
And there are a few enemies that need to be chain-stunned, but that's totally different and only appears in endgame raids, and if you don't know that going into an endgame raid like T5 or Leviathan Extreme, you didn't read up enough on the fight beforehand.
Oh, and one other thing - PLD stun doesn't become 6s until level 36. Before that (level 18-35), it's only 3s. DRG Leg Sweep is 3s at all levels, so at level 35 and below, DRG stun is way better than PLD stun (off GCD, no TP cost, 20 potency higher damage). DRG Leg Sweep just gets a CD decrease at 28 from 30s to 20s. Spineshatter Dive (level 40) though is only a 2s stun. But, Brayflox NM is level-synced to 34. So, why was that PLD complaining that you were using a superior ability, preventing him from using a far worse one? Oh, right, he was a moron.