You ignore them, do not mention such things on those types of forums and then you give up on them and come here where we're cool with both!
=D
A parrot will smell far less, if at all. Rabbit urine is strong, they spray, cage bedding will need to be changes a ton. Initial cost of a bird can be high, but the rest is cheap if your bird stays healthy. A rabbits initial cost can be low to free, caging can be also very low, but the bedding tends to add up. A vet for both will get pricey. You can use towels in the tray/cage instead of pine shreds, it will cut costs way down, but you will need dedicated towels and wash them separately.
Both need time out of their cages to run/fly around. Bird food can be a pain, since they are designed to waste a ton of it and eat little. This is my biggest issue with our bird, I absolutely hate things that waste food! But I still like him somehow...
Birds like to sing and chat, if you spoil them or end up training a bad behavior somehow, they will screech on and on. I discovered our guy is quiet most of the time when he can be on the deck or in the dog room. He can see all the animals and the big world, so he sits and watches them, sometimes singing or warning off wild birds that land on the deck. =D If he is in the house, he has to be out of his cage...=(
A rabbit will chew up everything, you need to baby-proof the house and then keep it clean. Similar baby proofing with the bird, but in a different way, fans, fish tanks, being sat on/squished type issues.
But if noise is an issue, go with the rabbit. Even if you move, its easy to hide a rabbit if you keep up the cleaning. I would never have one in the house again [pee smell], but that's just me.