Well, parrots don't actually speak, they mimic. They can be taught to say really interesitng things, but the parrot itself doesn't know what it's saying.
I'm guessing that all species of life would have the ability to utilize thought-speak, they simply haven't reached the level of evolution that allows themt o access that aprt of their brain. So, Andalites are evovled enoguh to use it as the main form of communication, while humans still ahve to use our mouths.
As for when in morph...it's possible that the morphing process allows the brain of the morphed animal to utilize that area of communication. I don't know how it would work, though. Perhaps morphing technology works using nano-technology.