Just curious - is it a zoning problem, or a homeowners' association?
To show that it doesn't make sense to ban ducks as pets, I would start by comparing them to pets that are allowed.
Dogs can be dangerous to your neighbors and cats are dangerous to wildlife, but ducks are a danger to no one.
Dog urine can damage landscaping plants and grass, but ducks eat slugs and caterpillars that are damaging to plants.
Dogs bark and howl and can be a noise nuisance, but even the loudest duck is much, much quieter than a barking dog. Ducks usually only quack when they are separated from one another. And their quack just sounds like natural wildlife. (My neighbors love hearing my ducks.)
Dog and cat feces attracts flies, stinks, and is can be a real problem for neighbors. Duck feces is small and loose and has very little odor, (as long as it isn't piling up in a coop). You could even bring in a duck poop so they can see for themselves?
The community board probably has concerns about people farming animals within the neighborhood. They can allow ducks as pets without allowing farming by limiting the number of ducks allowed to, I would suggest between 3 and 6 ducks. I live just outside of Portland, OR. The legal limit in Portland city limits, I hear, is 3, and at my house it is 6 (you can get a permit for more if your property is big enough) and that seems to be working. They can ban noisy animals like roosters.
You could also bring in articles about people keeping ducks as pets within cities.
I would focus the argument on the fact that your ducks are beloved pets, just like a cockatiel or parakeet or any other bird that they do allow, and that there is nothing about ducks that makes them any less appropriate to keep as a pet in your neighborhood than dogs or cats. If you have pictures of your ducks being held or otherwise looking pet-like rather than farm-animal-like, take that. I think I would actually avoid talking about egg production, and keep the conversation about them being pets. They care about the neighborhood - they don't care how fresh your eggs are. Suggest rules and limits they could impose to address any concerns they have about housing, odors, noise, etc., so they don't have to ban ducks altogether.
Good luck!!!