Truman, my Cape Parrot, learned the flighted retrieve in no time. Being a bird of the air that he is, he made the transition from running around (if you can call it that) to get the ball to flying for it without any special thought. You see, unlike Kili, Truman has never been clipped in his life and is completely at ease in the air as any bird should be.
Normally I develop videos and training guides for new tricks that Truman learns but this time there was really nothing to show. One time I rolled the ball on the floor and Truman walked back to me with it, the next time I kneeled with my arm higher so he flew up to it still holding the ball when I called his name. That and a few sessions of practice was all it took before Truman would fly all the way across the room to any place on the floor to fetch the ball and bring it to me.
I even took the trick a step further and taught him to land on my hand and then walk over to my shoulder with the ball before giving it to me. This too was extremely simple to teach. First I had him give me the ball on my hand, then I started having him walk more up my arm to give it to my other outstretched hand. Eventually I stopped signaling him to walk to my shoulder and he just continued doing it out of habit from the previous trials.
So instead of going into further detail about how easy it was to teach Flighted Fetch to Truman, I will tell you how I taught it to my Senegal Parrot Kili which is more likely what others would need to go through to teach the trick to their parrot. Kili already knew how to fly and how to fetch, so it was a matter of making the connection in her little bird brain to do them simultaneously. So for obvious reasons, your bird needs to already be a veteran at fetch (and be able to do it on different surfaces with different objects with ease) and be capable of flight. Ideally the bird should be flight recall trained but it is technically possible to teach a parrot that doesn't normally fly to you to do fetch on its own anyhow.
I found a pair of flat/simple chairs best for this but you could just as well use two coffee tables or other surfaces of similar shape/size that can be placed up against each other at first. Start by having the parrot walk around on one surface fetching the object at one end and bringing it to a cup or your hand at the other end. Use the objects and retrieve methods already most familiar to the bird. Once the parrot is comfortable fetching on this surface, begin having it fetch back and forth between the two joined chairs or surfaces by walking across the gap. As the parrot becomes better at fetching across the divide, slowly begin to spread the distance between the chairs with each subsequent retrieve.
At first the parrot will walk across the short gap, but as the gap widens it will have to hop and eventually flap to make it across the divide. By the time your parrot is flying from chair to chair to deposit the ball, you'll be able to place them quite far apart and the parrot out of habit will continue to fly between them to fetch the ball. All that is left now is to teach the parrot to use flight in other ways such as down to the floor and up to your hand with the object. At least the basic skill and concept of flying with an object has been achieved.
You can go back to the twin chair method but substitute the second chair for your hand. Place your hand within step up distance of the only chair you are now using. Ask the parrot to fetch but instead of flying to the other table, get it to step up on one hand and then deposit the object into your other. Just like the prior process of spreading tables, move your hand further and further away. You may or may not need to use your parrots flight recall cue to motivate or hint it to fly to your hand once it has retrieved the object.
Some things to keep in mind for teaching this trick. First of all, this is a fairly advanced trick. Don't expect your parrot to learn this fresh out of learning retrieve or its first few tricks. I waited two years since Truman learned to fetch before I decided to add the flighted element to it. I wanted a long term rock solid history of retrieve beforehand. The most common difficulty will be the parrot dropping the object when it tries to fly to you or forgetting to bring it and flying to you. The best solution is to use my spreading method because then the gap grows so gradually that the parrot transitions from walking retrieve to flighted retrieve without realizing the mobility change. If you try to go straight from normal retrieve to flying up from the floor with it, the parrot is likely to be confused. If you encounter any trouble at any stage, just go back to the last success point and keep practicing. Make even slower progress before increasing the gap. Remember to practice and perfect short distance flighted retrieves before moving onto longer ones. Once the parrot grasps the concept, be sure to challenge it and you will be surprised at how extremely capable they really are!
You can even take this a step further and incorporate flight into all tricks that are derivatives of the retrieve. For more fun and greater exercise, I have Kili fly recalls across the room for the opportunity to perform a trick. Then I create a divide that mandates flight. So instead of laying out coins on a table for her to put in her piggy bank, I throw the coins on the floor and then Kili flies up and down with them. Likewise with ring toss, I toss the rings on the floor so not only is Kili challenged by the colors puzzle but also to physically go and get them.