I think the "looking with one eye" behavior is related to what part of their eye has the best visual acuity. For example...hawks will turn their heads upside down in order to watch something overhead. That's because the part of their retina with the most sensitivity is located in the upper area of the eye (which enables them to see prey far below them when flying normally). When they have to watch something above them, they turn their heads upside down to get the best image.
The eyes of non-predatory birds tend to be located more on the sides of their heads so they have a broader field of vision all around their sides and back (to spot a predator better). Their visual acuity is probably better directly to the side, so using one eye may be better than trying to focus two eyes to the front.
For a cham, that has very mobile eyes that don't require much head movement to scan the world (and less head movements means they don't give their location away in foliage), each eye has two lenses and pretty independent visual acuity. If they are simply watching something, they don't need much precise distance focus, so one eye works, and the other eye can continue to watch their surroundings.
And, remember that what may have attracted the attention of your parrot or your cham might be the movement of the jet, not the exact image of what was moving. Most wild creatures' eyes are better at seeing movement rather than stationary objects. Predators and potential prey move...permanent features of their surroundings that may not be a threat don't.
