Parrots normally don't show any obvious signs of being ill until it's too late.
Hence, it's important to really know your bird, inside and out, so that you can pick up on any anomaly, as little as it might be.
I have a routine when I come home and greet the birds. To some I say an extensive hello (we "chat" a bit), to Joey I sing (he likes it when I sing to him) and with Petey...well with Petey I wait until whatever beating he's giving on of his toys ceases (beating I think which is meant for me...seriously, if Bram goes in the room first he'll get greeted with a nice "Hi Petey", NOT angry bashing of a toy) and then I softly talk to him for a few minutes. While I do all of this with them still in their cages, I observe them. Is there a change in their physical appearance (feathers, eyes, beak, etc)? Is there a change in their posture? Is there a change in the way they act? This is the time I take to make sure they are still perfectly healthy. In the past I have noticed some of them being off and depending on what it was, either I brought them to the vet or I kept observing them.
Two days ago, while doing this routine, I noticed that Joey's eyes looked a bit different. Not much, but enough to know it wasn't normal. I called the vet shortly after and got an appointment for the next day. Whatever it was didn't look like it was an emergency but it was enough to worry me so I figured going to the vet and get a clear conscience was the best course of action.
Turns out I was right. He had a mild case of conjunctivitis. In fact, it was so much in the early stage that my vet was surprised I caught it and did praise my sense of observation. Since it was caught so early on, just a few days of treatment (a drop in the eye twice a day for up to 5 days) should be enough. Just having the vet flush out his eyes and then apply a drop made a world of difference; by the time we got home, he looked much better.