Did pirates actually have trained parents that rode around on their shoulders? Where did this practice originate? If they didnt how did the myth originate?

You had a typo, you meant "parrots" and not parents.

Anyway, check out our FAQ, which has already answered that question here. In particular, check out the numerous topics under the "The Big Common Pirate Myths/Media Portrayals Roundup" where there several links to several questions (including a couple that answer the parrot thing).