Hi FSP,
What do you think about putting a pic on the webpage that is out of the ordinary (passport style). For example, would it be "professional" to put a pic on a webpage that would depict a person playing with a bat or holding a parrot or something?
No on the bat, yes on the parrot.
Actually, it probably depends on the culture of your field and/or department. I don't think anyone in any(?) field would be denied a job or tenure because their webpage photo shows them playing with a bat, but if this person already has a problem being taken seriously in their profession/department and/or has a department/disciplinary culture that values formality and a traditional view of professionalism, not to mention personnel who do not have rabies, this might give one pause before posting the playful bat photo.
Even so, depending on what webpage you are talking about, it might be fine and even a very good thing. Some departments have a main page with the stiff passport-style photos of faculty, postdocs, grads etc., but then there are links to research group pages and/or individual webpages. The degree of formality may be less in these more personalized webpages, and I don't think that is unprofessional to have informal photos there as long as people visiting these other webpages can get the information they want/need.
That is, if the main reason people visit these more specialized pages is to find out about research, publications, and people and all they see is a picture of you kissing your parrot, that's not so good. If the pages are informative and a bit informal as well, that's fine. "Professional" webpages don't have to be dry and boring.
And you could always have a link to "X's personal webpage" and put your parrot photos there. You probably want to keep photos of you in your underwear for a (private) Facebook photo album or to send (via Twitter) to the US Representative of your district, but photos of you and cute animals (other than snakes) wouldn't be unprofessional if clearly on a page devoted to other aspects of your life. Not everyone wants such a page linked to their work-related webpages, but I've seen examples of this and did not think the person (student or professor) was unprofessional.
There are likely differences of opinion on this issue. A while ago when I was involved in updating the department webpages, there was one person who refused to provide a photo. For a while I just had "no photo" written in a box the size of everyone else's photo, but this was unsatisfying and ugly, so I decided to try to get some image related to this person. I asked him to give me some sort of image to use, even if not a photo of himself, or else I would come up with "something" for him. He did not provide an image, so I put in an image of my own choosing. He loved this image and was quite content with the situation, but a significant number of other people were bothered by it, saying it was "unprofessional". Eventually I did get a photo of the person in question and the problem was solved, but it interested (and surprised) me how many people commented on the "unprofessional" nature of including an image that was not a standard head-shot photo of a person.
So, I may not be the best person to as, but perhaps others can weigh in, ideally specifying their academic discipline so that we can see whether wacky photos are the norm in some fields but mildly to deeply unprofessional in others.