I have limited access to verify which attachable VFXs suffer from this problem, but at minimum the shoulder mounted parrot does. Using any of the listed emotes below will cause the placement of the VFX to break, relocating from the shoulder to hovering in front of the shoulder as per the attached screenshot.
The emotes that cause this are;
@emote 10 (taunt animation)
@emote 17 (knockdown backwards)
@emote 25 (pushups)
@emote 26 (dancing)
@emote 27 (holding hands behind back)
@emote 28 (sitting - legs drawn up)
@emote 29 (pushing)
@emote 30 (grapple)
@emote 31 (point)
@emote 32 (holding chin, hand on hip)
@emote 33 (cower)
@emote 34 (crossing arms)
@emote 35 (sleep in bed, left-aligned)
@emote 36 (sleep in bed, right-aligned)
@emote 37 (crouch)
@emote 38 (hanging from ceiling)
@emote 39 (attacking/splitting wood)
@emote 40 (sleep on floor)
By comparison, other wearable VFX such as the masks (included in above screenshot), are not impacted by this. Something about performing these emotes, most of which are the newer ones implemented relatively recently, changes the location of the parrot anchor point until the user relogs.