Iago is an interesting sidekick for Jafar, but ultimately falls into the category of a Dark Reflection.
His whole purpose was to act stupid and brainless in-front of the public, and then whisper sedition and mockery in the shadows. He was supposed to fly around as a friendly parrot who couldn't ever do anything wrong, always betraying their trust right under their noses.
Iago was a darker reflection of Jafar, a powerless version of his own personality. Jafar kept him around because he respected Iago's ability to remain incognito and 'hide his powerlevel', which was the very trait which allowed Jafar himself to rise so high in the world.
He also served as a conduit through which Jafar could punish himself, a whipping boy. When he hurts Iago, he is assaulting his own ego by proxy, and when Jafar listens in delight to Iago mock, he is only listening to his own voice given flesh.
Just do a quick search of all the images related to the two characters, nearly eighty percent of the time they mirror the other's facial expressions and body language.
Iago was truly the manifestation of Jafar's mind, the Dark Reflection.