Hello! Welcome to the forum!
Pete and Kiki welcome to the flock!
I do have hope for Pete!
( ok , wow I packed a lot in here! I just really want to help. Might have to read in stages lol )
Its frustrating to get new friends and have them be fearful i know . Keep researching and reading. Take in everyone's advice as there can be different approaches that accomplish the same goal .
First to make them more comfortable and have more options to move and feel secure. I would suggest adding a few more short fat perches..they don't have to go the length of the cage .
And to make training easier. I suggest attaching a perch on the outside of the cage they can step out to from the door. I like putting perches on the outside top and sides as well, the rope ones work good. This turns their cage into a play stands area as well.
I find it much easier to work with birds out of the cage as some can be cage protect. Quakers are known to be cafe protective.
If you do this , its best to shut doors to all other rooms, hsve blinds or curtains over windows, ceiling fan off, no other pets around. And give yourself plenty of time to fool around. Have treats, a single safflower seems to work great for both species @ ( I have Quakers and a green cheek myself , they love safflower seeds)
Now you are almost ready. We are going to try and prevent this.
But what happens if they panic and fly off? Remain calm, don't talk or chase. When they land, walk calmly to them and talk to them . They should be happy to be rescued and even ones not trained to step up often will. Tell them how brave and walk them back to the cage and place them on top. If the first time you start moving towards them they start to fly off again , stop wait, when they land again, try again to talk softly and walk over and scoop them up or have them step up. Same thing if they fly off while trying to walk them back to the cage. Its ok, they usually tire quickly, it often doesn't take many attempts to get them back to the top if cage. Its possible they could fear bite, but you will just have to ignore that and take the bite .
But what we would like is to have them come out if the cage calmly and explore and hang out maybe go in and out of the cage. I would pull up a comfy chair, snacks and beverage for you . Just sit and hang out for awhile . Then open the cage , and sit back talk to them but don't focus 100% on them. Let them decide when to come out. I've given it as much as 2 hours before, just waiting for them to get brave enough. You can hold a treat at the door and hand them to them if they come over. You can show them some treats and place on top of cage .
What I like is if they come out check stuff out and start getting comfortable. I keep just sitting and talking and handing treats out if I can . And reading my book. Then I try to get them to go in and out of cage by showing some millet and putting it just inside the cage, or a slice of apple. Then one of the times they go in I lock them back up. I repeat this several times over the next few days.
If you use the same words and phrases they will start picking that up. Like time to come out, time to go back. I've trained mine to return to the cage when asked and I always reward this with treats and lots of praise.
Also sometimes when I'm sitting all smoothed up to tge cage tgey will start climbing on me and explore. I give so many treats if they do this. You just have to adapt to how things are going with you guys.
Target training to come out to a perch can be very help too. But the first part if target training is to just have them touch the chop stick and get a treat. I will link some videos and articles
This one is great, shows how to reward fir allowing you to come near. Then how to first teach touching the target. Just a few repetition then breaks is key with parrots. Note this is one day miracle! Lol don't expect things to go that fast for you . Clickers are great, but I don't use them, I say good birdie as the " bridge" before the reward
