Limiting myself to species I've seen alive and in no particular order:
Budgerigar
Gorgeous little bird with a charming voice. A group them socializing is a joy to watch and hear. I prefer the wild type, but the "domestic" blue variant is also great.
Red-lored amazon
A very typical amazon parrot but the most beautiful in my opinion, with a gorgeous colour combination on the face and a completely green body.
Elegant parrot
A flock of this species is a gorgeous sight, and I'm unsure why zoos in Europe don't seem to like this species. The two shades of blue look perfect on the light green bird.
Golden parakeet
A bird I hardly believed could exists until I saw one at a parrot sanctuary. On pictures it looks unreal in but the colours fit perfectly on the bird in real life.
Red-shouldered macaw
A typical macaw but very small, this bird is both handsome and cute. Almost entirely green with a blue-ish crown and red shoulders, this species is not overly colourful but very elegant.
Thank you for the comment
@Mr. Zootycoon !
So this is a list comprised of species not yet mentioned, interesting.
I get what you mean about the budgie. I do think it is often underrated because of how common it is in captivity but I also find them to be an engaging little species and I like watching them (in captivity) too.
Red lored amazon parrots are beauties, I absolutely agree, the plumage is wonderful. I've seen these in the wild in Mexico and Central America and they are such charismatic birds and breathtaking to watch in flocks flying.
Elegant parrot is an interesting choice, it doesn't have the most vibrant plumage compared to many other parrots but as the name suggests it does look very "elegant" and distinguished

. I don't think that I've ever seen this species, have you seen these in the wild ? or in captivity ?
Golden parakeet, you beat me to it, lol, this is probably my favourite parrot of them all, will explain why in a future comment with my own list of top five parrots.
Red shouldered macaw, yes again I agree and in fact for me I do love the larger and more vibrantly coloured macaws but is the smaller and mid sized ones which perhaps interest me much more.
1. Spix Macaw- because I waited near enough fifty years to see this almost mythical species.
2. Kakapo. Amazing bird. Never expected to see one but I've even held one too!
3. Military/Buffon's Macaw. Like the more subtle colouration compared to the more brightly- coloured Macaws.
4. Major Mitchell Cockatoo. Favourite cockatoo- beautiful colours- seen in the wild too.
5. Hawk-headed Parrot- saw it described as a 'bizarre beauty' once, which it is. Haven't seen one in many years though.
Extra; Imperial Amazon- this is one I probably won't see now, due to rarity in captivity, there is one pair at ACPT in Germany I believe, and a single handraised one in Florida(?) which is the offspring of one of the ACPT birds from before they left Dominica. I may have seen the one at ZSL in the distant past but cannot remember it.
Thanks for the comment
@Pertinax!
Have to ask, where did you see a kakapo ? In the wild or captivity?
You held one too ?! That is incredible ! , are we talking about a live kakapo or was it a taxidermy specimen ?
Focusing on the other species that you've mentioned that haven't yet appeared on other peoples lists.
The spix macaw is a very enigmatic species, I agree, also have a great significance for me too though I've never seen one yet (hope to one day though), where did you see one of these birds?
Millitary macaws are beautiful parrots, I've seen these in the wild in Mexico and they are truly spectacular to see in flight and I love the green and red plumage.
Hawkheaded parrots, these are interesting birds, I love that magnificent crest that they have though they tend to keep it tucked down most of the time and are really quite shy and retiring birds in captivity.