The parrot bit does rather raise some questions. Of course ballads are seldom logical but - Why did she tell the parrot ANYTHING? Why didn't she say, "Mind your own business, you snoopy parrot!" and leave it at that. At least the bird in the ballad "Young Hunting/Henry Lee" witnessed the murder. You'd want to offer a nice golden cage to that bird. The whole thing makes more sense in that story.
Also why did Polly call her pet parrot after herself? Was Polly - the girl, named after the pet parrot?
I do so enjoy these discussions. Where are they leading us? Is this the path to true enlightenment? If it is - can I keep up? Joy