I don't think Alex the parrot could have passed the Turing test, but his use of language was rather impressive. Given that such a conversation bot would be an extreme specialist, parrot intelligence is much more general, and the bot would be only slightly better at its specialty than Alex the parrot is, I'd say it's highly unlikely that such a conversation bot would be smarter than the average parrot (which I assume to be only slightly less intelligent than Alex) by any reasonable measure of intelligence. Same goes for dolphins, and probably even for labradors.