Interesting marketing and design. With the unit near flush with the ceiling it clearly can reduce the problem with a fan blade and a Parrot making mid-air contact. Although operating near flush to the ceiling (i.e.: there is little room between the unit and the ceiling). The body of the unit does turn and there is (although limited) the contact with the spinning body and the a wing catch between the unit and the ceiling. Remembering that the likelihood of that event due to the air flow away from the unit in that area would be low.
If you have an open concept floor plan, the unit will likely not be as effective because it needs the side wall to 'help' direct the air back down to the floor.
NOTE: The inflow at the center of the unit 'may' trap a smaller bird, however that is subjective since I could not find an in flow rate listed. That would be a question for the manufacturer.
Clearly placement is an issue. The nearer to the center of a room, the better overall effect. NOTE: The flow direction appears to favor a 'Winter' pattern, where air is lifted at the center of the room and flows down the outer walls/ a 'Summer' pattern reverses that flow with the air lifting at or near the outside walls and is pushed down at or near the center of the room. So, that becomes a personal preference thing!
With the limited problems compared to a traditional fan, it has merit. However, I would be careful with its use with smaller parrots and birds. Having said that, it is very possible that its air movement is so passive that it may not be a problem. If true it could take some period of time before one even feels the effect.