I have kept hamsters in my room when I was younger that my mom did not approve of. Hamsters are fairly easy to keep out of site for people that fear them. As for your Dad there are a few things that you can do, for dogs that is, I am with him on the bird thing
First aim for a smaller dog one that isn't too "yippy" with the barking, that limits you a little bit because small dogs tend to be a bit on the barky side. Stay clear of Chihuahuas and, believe it or not, Maltese, those are two breeds that are notorious for becoming very protective of their "master", and can get snappy even with other members of the family. But something like a miniature poodle, or another one, not sure of the exact breed, but they look like miniature border collies with floppy ears, they are overall very family friendly dogs, even with people that fear dogs.
Second, start to introduce him to dogs, friendly dogs, ones that you know won't be snappy or to barky with him.
Third, find out why he is afraid of dogs, you are never going to cure him of his phobia with out finding out the reason. In my experience most people are afraid of large dogs, due to an attack as a child. When I say attack, I don't necessarily mean a full on mauling, but maybe chased by a dog as a child, or lived close to a mean looking/sounding dog as a child. Most people don't fear small dogs as much, I guess they just don't look as menacing as an excited, jumping, barking German Sheppard.
Finally, if you are able to get him to agree to a dog, bring him with you when you pick it out. It is amazing how quickly people who are afraid of dogs can become attached to a cute little puppy.
I suppose the same method would work for birds as well. Most people who fear birds tend to do better with "smarter" (for lack of a better term) birds, such as parrots (a smaller breed in your case) or cockatoos, they just seem to have a little more personality then the basic finch or canary. Nothing against those birds it is just that a lot of people (at least that I have seen) find the "smarter" birds are more interactive and funner to be around. I just don't like them because I find them to be messy, well, more messy than a cat or a dog that does their business outside or in a litter box, and now they have self cleaning litter boxes
As for your Mom with the allergies, my husband has asthma and is allergic to both cats and dogs, but we have 5 cats and dog, he has also had pets growing up. Every pet he has ever had, he has become accustomed to that animals dander, and usually after 2-3 weeks his allergies stop bothering him. I know that is controversial, and most people would disagree with me on this, but having worked in several animal clinics and and in human allergy clinics I have seen it happen time and time again. People with allergies become used to the allergen, and eventually the allergy goes dormant (again lack of a better term) it will never fully go away, but I have seen it become less, if at all, bothersome to the person. It really depends on how bad your Mom's allergies are, ie. if she has anaphalaxis to cats, that is where she has so much swelling with in her throat and face area that she cannot breath, it would NOT be a good idea to try her with cats. But if it is just that she get hives, there is a possibility that she may become accustomed to the cat and find that the hives go away. But every person is different and if you and your Mom are willing to give a cat a go to see how she does, it should be done with extreme caution and some good anti-histamines close by.
As for you Mom with out liking animals with bare tails, I am TOTALLY with her on that ... stay away from them
I really don't know how to fix that, I have yet to overcome that phobia myself.
As for suggestions on what type you could get I don't really have any experience with reptiles, so I can't help you there. But have you thought of a Ferret, or a prairie dog. I think technically they are part of the rodent family, but not as "rodent like" in behaviour and looks, they are a little on the pricey side, but I have heard they can be fun pets (an no bear tails). You might be able to pass with the hamster, as they have no bear tails, actually hardly any tail at all, but furry none the less, and they aren't a stereotypical rodent, they are a Cricetinae which is more closely related to the Beaver, which is a Castor, a type of rodent, but most people don't think of beavers as rodents, it might work with your Dad.
Plus hamsters don't really look "rodent like", and if you get a long haired hamster they really don't look like the typical rodent.
Good Luck
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