I'd vote for cat only because you have kids. A kitten would make a better pet.
However, I also think bunnies are underrated. They're great because they're ADORABLE, and will eat just about anything, and kids tend to get a kick out of being able to feed their animal, ime cats don't really take treats all that well. It takes more preparation since you need to bunny proof your house, however bunnies are the 3rd most popular pet, and just as overpopulated. I raise meaties, but I still think they deserve a bit more recognition. They're overpopulated as well, and can really make great pets, though they're a bit more boring. When you get them really comfortable, they can be super cute.
Overall, here's the good / bad aspects.
Buns/good:
Sweet, loving
The bunny your looking at is cuddly. Most buns are fine with being pet, but cuddle doesn't suit them.
Overpopulated as well.
Softer than a cat (IMO)
They binky! Ever seen a rabbit binky? Oh, it's too cute!
More willing to eat from the kids, something I've found kids love
Can be litter trained (better when they're spayed)
Won't eat parrots. May attack a parrot if felt threatened, but highly, HIGHLY doubt you'll find a bunny with a taste for parrot
Sharp nails
Buns/bad:
Smellier overall
More proofing of the house
Chew on things (but when you own parrots, its not much of a difference)
Cost more to feed? (They are fed pellets with hay, and veggies As treats)
Not completely litter box proof, but any stray poops are easy to clean, and stray pee isn't that bad
Kittens/good:
Black cats are THE most unwanted cat
Cuddlier
Smell less overall (but the litter box STANKS)
Cute/ sweet
Loving
Litter box trained well!
Have use over their nails
Purrs
Kittens/bad:
VERY sharp claws when used
smelly litter box/smellier feces
Don't binky....
Will attack your birds if not trained properly / kept away. Even a lick could cause issues.
Kittens tend to find homes sooner than rabbits (and rabbits are neglected more often due to mis-information)
Choose wisely, but I'm kinda rooting for the rabbit... Kittens tend to get adopted fairly fast, even if black
Karen: I LOVE ferrets. Used to have a pair when I was younger. Got some last year. Not the same mistake ill make again, not until they have an ENTIRE room they can live in - AND NOT get out of. They killed several chicks, and a pigeon

we got rid of them after the pigeon. I loved that pigeon! I still love ferrets, but I want an entire room they live in, can play in, and not get out. 2 doors - the door to the room, and then a door you open after going into the room. Then ill own more. I really do love them, but gaah! They're such predators. I tried to feed them a chick they killed, since raw meat is good for them.. They just looked at it, confused. We fed them some mice. It was HORRIBLE. I don't ever want to watch a ferret eat a mouse again... The mouse was dead, but... Eew.