My parrot is flapping wings in his cage. Is he bored or exercising?

He may simply be playing and having fun. If he is doing it all the time, he may be stressed or bored. Try rearranging the perches and switching old toys for new ones to freshen up his environment. Spend more time with him outside of the cage if possible. When he exhibits the behavior distract him with toys or treats then reward and praise him when he responds to you.

Without knowing the size of the cage, it is hard to give advice. If they have a large flight cage where they can easily move around and exercise, then that may be enough. Adding in some fun toys to explore, moving dishes and toys around or rotating them out, offering treats to chew on, etc can all help increase enrichment and make things more fun for your birds. If they're kept in very small cages, then looking for a playpen that can fit in the room or on top of the cage may be nice for your birds to climb on or play on. However, finches tend to explore more and may want to fly around, so it is best to make sure they are only let loose in rooms where they can easily be caught again and cannot fly into windows, doors, or other dangerous objects that can cause injury or death.

He is bored. Provide a variety of toys and puzzle feeders for him. Increase play time, both in the cage and out, to keep him distracted. When the behavior is seen, a stern "No" should stop him and then he can be distracted. Be firm and be consistent.

It might be that John is just very excited/happy in that moment and is wanting to let everyone know. If he's otherwise acting healthy and normal I wouldn't be too worried. You can try offering a treat or placing a new toy in his cage to see if it helps keep him happy and interested in something to play with.

This can be normal parrot behavior. They may flap and vocalize if happy, or communicating with thier "flock" which may be their human. They may also vocalize as mating behavior. If this doesn't answer your question please repost. Here's a little more info on normal bird behavior. https://lafeber.com/pet-birds/bird-behavior/

Congratulations on your new birds! If the flying back and forth is in response to when you are putting your hand into the cage this could be due to fear or being startled. If he is randomly flying throughout the day with no relation to you approaching then it could be him just exercising. Millett is a treat that is very useful in helping gain your budgie's favor as well, so letting them nibble some from your hand or while you touch them if they are comfortable with you touching is helpful in making interaction with you a pleasant experience and can eventually be used to lure them onto your finger or hand!

He sounds like a depressed parrot. you need to let him out of the cage every day, you need to put his cage in the living room, ideally close to a window view. you need to start feeding him a decent diet (no more seeds, only as treats, he needs to eat parrot pellets and fruits mainly). once you start doing all these things he will hopefully get much better.