I was just going to post to try using a dowel type perch! Let him step up on that and then take him out of cage, put him on table or arm and let him explore. Awesome you tried the pen!! Great option! And sometimes green cheeks can get a bit cage possessive, so having Parry trained to step up on a perch may save your fingers when he/she goes thru puberty / hormonal phase.
He sounds like he is very happy and loves interacting with you. Heading bobbing and back and forth in reaction to you is kinda like a puppy dancing around all excited. He's coming into his own and learning about his new flock
I am guessing his wings are probably clipped so he may not fly very well. If he flutters off or gets too excited you can offer him the pen or perch to step up on if he won't come to your hand or arm. Absolute last resort is to cover him with a light kitchen towel and hold him firmly around the body to get him back to cage. It won't hurt him. And the towel isn't terrible, first couple times he might be upset with you, but he won't hold a grudge much! Getting him used to the towel will be really helpful in the future for trimming nails and if you do decide to keep wings clipped. But he seems so interested in you, I doubt he'll be flying off, he'll do what my daughters little pineapple green cheek girl does, she follows Victoria from room to room to check out what shes doing.
Congrats on the baby!! And don't worry about buying a bird cause you feel sorry for it. Tho that may be the way it seems to be described, I am sure you've seen lots of pets that you felt sorry for but did not bring home. I am guessing there was some other pull to Parry and he was just meant to be in your family!!
A note about having rats in the same home... rats are meat eaters. Be very careful that Parry never lands on the rats cage. Even the sweetest of rats (we have shared our home with many fancy rats over the last 10 years or so, none now tho)... but, we had an incident when the cockatiel babies first learned to fly and one landed on the rats cage, before we could get there, one of the girls bite the tip of a toe. Poor Rin still doesn't have a normal claw on that toe. Rats are super awesome puppy dog kinda pets, so would never suggest they cannot be in the same home, but just be careful with where Parry explores or throw a cover over the rats cage when he's out. We used that option quite a bit!
Good luck and post more pics!! Love seeing them as they find their inner bird
Congrats again and so happy that Parry has a wonderful family to call his own!!! Rats and all!!