Welcome to the community! I think you'd and Kelly both would be wise to stick around here, post often about your lives with Enzo, and to find support with other parronts who are going through the exact same thing that you guys are...It's extremely important for both of you to know that what you are going through is extremely normal and common, you're not at all alone in your situation, and that there are answers, help, and compromises that will make your lives with your CAG much less stressful, and eventually and hopefully a loving household...And you re-homing Enzo after you've had him for over a year and after you have both bonded to one another should never be an option, as there are lots of things that you can do to at least help, if not eventually remedy what you're going through...
I understand where Kelly is coming from, and it is very difficult when you feel jealous of a child, a pet, or a friend of your significant other. However, if Enzo was your human-child, I'm sure Kelly would not ever suggest or even think about you "re-homing" him, lol, and that's what Kelly needs to really come to terms with, that Enzo, being a young CAG, really and truly is the equivalent of a 4-5 year-old human child, and as=such, he is going to require much attention and time. So hopefully this little blow-out that you guys all had was an eye-opener for everyone that re-homing Enzo is not at all an option, and that the only way to figure all of this out is to take it slowly, realize this is going to be a marathon, not a sprint, and that there are many, many things you guys can try that you haven't yet in order to get Kelly and Enzo to better accept one another, and hopefully become friends.
At the same time, you need to realize that Kelly isn't a "parront" or a parrot person, or at least it doesn't sound like she had much experience with parrots prior to dating you, and for people who have never owned a parrot themselves, they often look at a parrot as just that...it's a bird. It's a pet. It's no different than owning a dog or a cat, or a ferret or a rabbit. From Kelly's point of view, at least in the beginning of your relationship, Enzo was just your pet bird, nothing else. And at this point even a year later, Kelly still has not ever in her life formed a close-bond to a bird herself, so she still probably does not have any concept of what that is like. I don't know if Kelly has any human children, but it's important that you impress upon her that there really is no difference between having a human toddler and having a parrot, especially a CAG, who literally has the intelligence of a 4-5 year-old human child. So she needs to learn that Enzo is not a dog, he's not a cat, he's your child, and also that he's a "perpetual child" that is going to live for many decades, just like a human child would. Once Kelly gets a better understanding of your relationship with Enzo she will definitely be more understanding of not only the amount of time that you absolutely must spend with Enzo, but also with the fact that she can't give you ultimatums regarding Enzo...But it's going to be your job to teach Kelly all about parrots, take her places to try to interact with parrots, do reading about them, and most of all, it's your responsibility to ensure that the 3 of you, including Enzo, do things together as a family on a regular basis. I don't know how often you take Enzo out and about or if he's harness-trained, but if he's not I highly suggest that you buy him an Aviator Harness and get working with him on a daily basis so that he learns to accept it, and that the 3 of you start doing things together as a family, even if at first that means simply taking long walks around the neighborhood, hiking, etc. with Enzo on your shoulder, the 2 of them need to start spending family-time together with you (usually when they realize that wearing the Aviator Harness means going bye-byes, outside or in the car, and going places like pet shops, parks, hiking, shopping at stores that allow pets like Lowes, Home Depot, Tractor Supply, Walmart, etc., and they directly-relate the Aviator Harness with outside, they willing accept it)...Kelly obviously hasn't yet formed any type of bond with Enzo, and Enzo may or may not care for Kelly (hard to tell as his jealousy and hormonal behavior regarding you as "his" is currently getting in the way; sometimes birds actually very much like and become attached to people that they see on a regular basis and who don't hurt them, even though they outwardly are aggressive towards them due to jealousy, hormones, etc)...
Getting the 3 of you to spend more and more time together on a regular basis, in situations that are not stressful as they have been so-far could help their relationship quite a bit...Kelly and Enzo don't necessarily need to directly interact with each other, meaning "touch" each other, lol, obviously that isn't something you want to do when it's a safety concern...But with you showing Enzo that it's very possible for Kelly to be included in your time with him, and that Kelly being there with the two of you does not mean "less time for Enzo" with you, the sooner Enzo is going to accept Kelly into his house with his person...And that's on you my friend, instead of playing "mediator" as you have been so-far between Enzo and Kelly, you need to teach Enzo that it's okay for Kelly to be with you, because Kelly being with you does not in any way take you away from him.
You nailed-it right on the head when you stated that it's not Enzo's responsibility to be the "adult" in this situation. CAG's are sooooo intelligent, I'd go out on a limb and say that they are the most intelligent of all parrots (I grew-up with my mom's CAG, who she raised from a 3 month-old baby, and who she still has, he's now 32 years old and he really has been and still is my only sibling, he's my brother, and that guy is fully aware of everything that is going on around him, near him, and with everyone in the house, no doubt about it)...So it's kind of a double-edged-sword with CAG's, especially one who is still a baby at 14 years old, and who has already had multiple homes in his very short life...
Enzo has yet to be given one single, solitary reason to trust any human at all, and that's the fault of the people who have been responsible for him, not his...And the irony of this is of course the fact that none of these people have ever been "responsible" for him at all, even though they were in-fact "responsible" for him...Except for you at this point...However, you've only been his daddy for about a year, and something that you need to keep in-mind is that at this point, Enzo probably doesn't even fully trust you...And I'm emphasizing fully because he obviously does trust you, he's formed an extremely close bond with you, he loves you very much, and he's been able to overcome any of his prior "baggage", his insecurities, his discomfort and anxiety with people, and he's definitely taken a huge step in allowing you to be his "person"...That all being said, after only a year being with you, and after 13 years of being neglected and traded and sold very much like a used car, Enzo may still not entirely be certain in his own head that you are truly going to be his "Forever-Person" or his "Forever-Home"...And who can blame him for that doubt, after all, it's been created by the very people who were supposed to love him and keep him safe, secure, and be his family...Try imagining being 14 years old and already having had 3 "families"...If you really try to imagine that, it's quite unsettling and disturbing.
So what I'm getting at is the fact that you're still building your own very close, special bond with Enzo, and you're still working on earning his full-trust at this point...and it's your responsibility to not only ensure that you keep building this bond with Enzo and earning his trust, and making sure that you don't start to spend much less time with him because of your significant other, but also your responsibility to educate Kelly about just how intelligent, emotional, sensitive, and human-like Enzo actually is, about his prior history up until the point that you brought him into your life, and how absolutely devastating it would be to Enzo's physical, psychological, and emotional health if you were to drastically reduce the amount of attention you give him and the time that you spend with him...let alone if you were to actually re-home him for a 4th time in 14 years; that would be very-literally life-altering for Enzo, meaning the rest of his very long life would be effected in every possible way negatively, and it could very well devastate his overall quality of life forever...I don't think there is any chance of that happening based on what you've already stated, thank God, but you need to make sure that Kelly fully understands the situation, and the dynamics of Enzo and your relationship and time spent with him...Hopefully Kelly moving-in with you and Enzo will be a very positive thing, and actually a help in the not only the 3 of you starting to spend time together, as a family-unit, but also will reduce the amount of time you'll be spending away from Enzo, which will in-turn lessen his jealousy towards Kelly....
I do realise that some peoples situation changes in life and in extreme cases birds need to be re-homed but still it must be so destabilising (correct spelling 
) for the birds as well as the parronts
