Parrots apparently have special voiceboxes compared to humans. In fact, rather than a larynx, they have something else called a syrinx. It's almost like a double voicebox which allows them to talk continuously through one while breathing through the other. Humans, on the other hand, have to stop to take breaths every now and then during their speech.
If they made it that Jafar found the lamp at some point and wanted to make a wish, but then Iago accidentally spoke a wish instead, that could explain why he is able to talk. If Iago said (and he was just mimicking) "I wish I could talk", then he could be granted the true power of speech via the Genie. That would be a quick and easy explanation for why he is able to communicate so eloquently.