I have roosters also why stop at just one...
Seriously yes you can make a pet of a rooster. I have a few that my DW will not let me cull or sell. They have other chickens but they do not have mates because they are not the alpha rooster. Depending on the roo you can make a pet of them but you must give them attention.
Chickens or at least mine in the daytime freerange and some are loners, but of a night they need to be around others. But then too we have hens and roos that like to roost alone.
One in particular is a white laced red cornish bantam who would rather follow me around then play with his chicken buddies. He used to be waiting at the back door every night for my wife to let him in where he would casually walk to the bedroom and hop on his parrot perch or sit on a towel covered pillow to watch tv until being put on the perch. I finally put a stop to it hoping he would bond with the other chickens. No such luck, he finds a spot by himself and sleeps. If I open the back door before it is completely dark he makes a beeline for the back door in hopes of getting in. When he was inside he did not crow until the alarm clock went off, my DW has to get up at 3am so the crowing was not a problem, I sleep right through it. Though he has been exiled from the house I do pick him up whenever I can to let him know he has not been forgotten.
So yes you can make a pet of a rooster, make sure it is a small friendly quiet breed. You have to have a perch with a dropping tray just like a parrot. You will also need a safe place outside in the daytime for him to forage, like a tractor or fenced pen. From what I understand there are some city slicker apartment dwellers who keep chickens inside, but I think anything bigger than a serama or a OEGB would be unacceptable.
Chickens have some advantages over parrots. They do not bite off fingers or take out chunks of meat when mad. The crowing is nothing compared to the screams of cockatoos or macaws at sunrise and sunset. And both the male and female parrots scream. Chickens can not chew apart furniture. Parrots like to get attention when they want it, much like a cat. Chickens will usually take it whenever they can get it, if they have been handled consistently. They both poop though the chicken poop may smell more. Parrot food is more expensive than chicken food, and parrots are WAY more expensive then chickens. Chickens do not talk but they do purr. Both chickens and parrots will eat your jewelry, though the chickens do not usually rip it from your ears. Chickens have much shorter lifespan, but then some parrots outlive their owners.
Ohh and chickens don't repeat things you say that should not be repeated...