Although parrots can be trained to say about 200 words and short phrases, as a species, they are not the best of the bird mimics. In fact, the top word mimic is actually the Mynah bird. Its ability to create amazingly realistic human and non human sounds such as words, coughs, sneezes and even cell phone ring tones is well documented.
Other birds that are extremely intelligent and known to mimic human words are crows, ravens and jackdaws. Although their abilities are less publicized.