Just wondering if such a feature has been added. In FFXIV some non-combat pets can be commanded with the emote /beckon to sit on your shoulder or head.
I’m asking this because I managed to get the parrot from Siren Isles and was wondering if I could get it to sit on my shoulder.
What I could find. 
There’s only really two:
The parrot that sits on your shoulders, and the sloth that climbs on your back.
And there’s cats that’ll sit in your lap, if you target them and do /sit.
But that’s it, really.
Pepe on your head.
I had to read that twice for obvious reasons.
There’s also the mechanical one, feathers I think it is called
There’s the sloth that will hang on your back.
The dangers of Pepe is he can popo all over your hair dodo, making it soso sticky and you know thats a nono waiting to happen!
My fave. Daisy. 
Accidental hair gel
Would make Acrobatic Stewards more interesting.
the two parrots linked above can sit on your shoulder, some of the cats will sit in your lap if you target them and do a sit emote, and i believe the sloth pet from the bnet store climbs up onto your back
lol so it’s been another long day and I’m tired one more hour and I get to go home but I click in here because I glanced at the title and saw the letter h in sit
lol “what?! Do that in your shoulder”
Sorry
I saw one about pepe on your head. Dont feel bad
It isn’t a battle pet but…
Lasts for 10 minutes.
Disgusting oozeling on my shoulder? Ew! Hes for critters to melt!
I think the variety of pets could be a problem for that.
More likely, you may get a shoulder transmog that gives you a parrot companion at some point.
They can’t even get shoulder armor to sit on your shoulders without floating most of the time.
Thanks for all the information guys.
Tried /whistle on my new parrot. It did nothing. 
Well, they have managed to get it to working for a few non-combat pets so the tech is already there. Should be trivial to get it working for other non-combat pets.