I've known several people who regularly brought their pets (dog/cat/other) with them for fieldwork. It's not something I would do, but it can work out under a few conditions:
1. The animal in question is docile/well trained and likes to be right next to/perch on the shoulder of its owner. No bolting!
2. The animal likes travel, or at least is easygoing enough that it doesn't care one way or the other. If it would be happier/more secure at home, then don't drag it around!
2. The fieldwork doesn't involve any heavy equipment or other major mechanical hazards.
3. The site is either secluded or contained in some way (i.e. not dispersed throughout a neighborhood or industrial area).
4. The presence of a pet is not going to ruffle any client or stakeholder feathers.
5. Nobody in the field crew is allergic to/petrified of the animal. Usually this works out ok if the pet owner has his/her own vehicle so that nobody else is subject to hair/dander/feathers/loose critters. Or if the coworker honestly doesn't mind. I used to spend long (6 hour +) drives with a coworker's animal, but I don't have allergies and I like the buggers so it worked out ok. Other people were seriously put out by it and it was a big black mark against them in the office/in their career.
I have had plenty of sites where a leashed dog/shoulder loving parrot/trailer hamster would be fine. But pet owners should keep in mind that an on-site pet may be viewed as charming/totally ok by most people, but that a certain percentage of the population will find it incredibly unprofessional. It's best to be extremely careful and to take into account political considerations if you want to take your pet along.