Perhaps it depends on the personalities of the specific animal and specific person, but my bond with my cockatiel is the strongest human-animal bond I've ever experienced, and I've lived with plenty of furry friends throughout my life. While I do often find myself wishing he was bigger so I could hug him, I pet him all the time. He loves it when I scratch his head. He may not run up to the door when I come from a long day like a cat or dog might, but he does start singing and chattering, which I find much more pleasant.
If you'd asked me before I adopted my cockatiel which animals bonded best with humans, I'd have agreed that the answer was cats and dogs, without a doubt. However, living with this little guy the past couple of years has taught me that parrots also belong on that list.
