What are "parsley greens"?
In most contexts, I would assume that simply means the leaves from the herb parsley.

Depending on where your recipe originates&mdash;especially central Europe, or some Asian cultures&mdash;parsley root may also be used, so the recipe might be trying to make that clear.

I suspect, however, it really is the recipe author's idiosyncratic style.

Now, as to chopping them, that would depend on how it is used and what the rest of the recipe says.