Trap/mine de-fusing. They are too light to trip most traps, and they can fly over triggers to get to the best spot to defuse stuff.
Rear attack. Obviously it's wonderful to attack your enemy from multiple sides simultaneously, but often it's difficult to 1) get troops in a separate position safely, 2) get troops out of that position safely. Parrots can fly. Even though they are not as strong as humans, being attacked from behind by a parrot is not something you can just ignore, especially if you are also trying to fire your weapon forward somewhat accurately at the same time. (Your enemy will start fortifying their emplacements against this kind of attack, but that's more time they have to spend on something that doesn't directly advance their position, which is still good for you. And if they try to skip that, deploy the winged forces.)
Not just forward scouting, but also sentry duty. It's going to be really hard to stay hidden from sentries who are flying.
Balloonist. Obviously parrots have a hard time carrying something heavy like a large shell full of explosives, and balloons are unreliable in that they tend to go in random directions. But if you harness up a team of parrots to a balloon, they can drag some heavy object out over the enemy, pop the balloon, and give the enemy the middle wing-tip as they fly away. (You might want some kind of safety harness where they can quickly release in case the balloon pops prematurely.)
Range-finder. They are good with wind somehow (not sure this has a factual basis, but we'll go with it) and they have good vision so they are good at figuring out what adjustments to make to artillery in order to actually hit the target. Plus, they can fly, so they can get a much better idea of where it's actually hitting and yell down things like "half a degree to the left and 80 feet further out". (Depending on how early early-modern is, this could even just be ballista or something.)
Less obvious non-combat duties:
Medical. Of course you have parrots working in the field hospitals as regular physicians and nurses, but they're also really valuable as first aid medics on the battlefield. They are really fast, especially if there are debris on the ground in the way, lots of people have had their lives saved by a parrot with some bandages.
Camouflage. They have better color vision than humans, and they can see what they're working on from more angles, and they're more mobile so they can work with multiple groups, moving from place to place as needed. Most camouflage artists are parrots.
Clergy. They're not particularly more effective than humans at this, but they're also not any less effective. (And, as mentioned in another answer, they are much cheaper to feed/house/etc. than a human, so any task where you can swap 1-1 parrot to human, definitely use the parrots.)