Different parrots enjoy different activities. Cockatoos you can completely and totally love up on them anywhere on their body! It's not necessarily good to do (re: hormones) but in the wild they are very affectionate birds. A grass parakeet on the other hand you'd be lucky if you could give one scritches! They are not big on allopreening, so typically don't enjoy the kind of physical interaction that cockatoos, conures or other birds may enjoy.
Also, even within the same species, some birds enjoy more physical interaction than others. It varies.
But the main thing is, is if you touch them in the wrong way, this can lead to an overly sexed parrot who gets the wrong signals from you... i.e. touching under the wings - something that only another mate would do. I know many people love it when their parrots "love" them in a mate kind of situation, but it's not a healthy relationship. We can't sexually please a parrot, nor can we raise a family with them. Essentially, we need to figure out what kind of touching and petting is acceptable for our birds and what's going too far.
