Well, budgies, cockatiels, lovebirds, beebees, etc are all quiet parrots as long as their living conditions are right. Budgies do chatter all the time but, to me, it's a soft background sound which is quite pleasant. There are people who go crazy with it but I guess they are the kind of people who can't stand any kind of noise and I am not like that. Cockatiels are quiet if kept in pairs and at a strict solar schedule because the only time a male paired with a female vocalizes a bit more is during breeding season - and you do not hear a single peep out of the female. But, if they are kept hormonal, the male will vocalize all the time and, if you keep a single female or a single male, they will both vocalize all the time calling for a mate.
Personally, I love cockatiels and they are the species that I always recommend for first timers because they have all the 'parrot' quirks but few of the problems: they don't scream, they don't bite and they are not destructive. They are also beautiful and smart and are, by far, the sweetest tempered of all the parrot species. The males learn to whistle entire songs (there is a video here of a cockatiel whistling The Queen of the Night aria perfectly) and they can even learn to perform tricks (I don't really like parrots performing tricks but some people seem to want them for this purpose alone). They can even learn to talk although they are not considered 'talkers'... And they are not hard to take care of because they are good eaters (they don't like fruits much but they love their grains and greens and enjoy a number of veggies). I had a flock of over 30 of them living cage-free in the birdroom when I had my rescue and I always enjoyed them immensely.
Conures are great but they do require A LOT of one-on-one and they will scream, bite and even pluck if you don't give it to them. I have two GCCs, both female, and they are both quiet birds (much more Codee than Annie) but Codee is on me a minimum of 8 hours a day and Annie has a BFF (Epuish Caique) with whom she hangs out all the time that she is out of her cage.