Twitter slang about talking shit about someone behind their back.
At date, the stats publicly displayed on a Tweet include a counter for Quote Tweets (QTs) — Retweeting the Tweet with an added comment of your own — which displays an accurate total number of QTs.
If you click on the QT counter, Twitter will show you a list of QTs on the Tweet; however, it will only show you the QTs that are publicly visible. Tweets by locked accounts, including QTs, are not publicly visible, so they are registered by the public QT count but do not show up in the QT list.
If a Tweet has 1 or more QTs but they're all private, the QT list will have no entries, just "No Quote Tweets yet," topped by a drawing of a parrot which appears to be narrowing its eyes suspiciously (the titular parrot of shame).
The only easy way to access the QT list is to click on the QT counter, which only shows up if a post has QTs. As a result, if you see the parrot on one of your Tweets, it means someone is discussing your Tweet on a private account, which often means they have an opinion about you or what you said, but don't want you to know what it is.
Therefore, referring to the "parrot" or the "parrot of shame" is a way of saying you know people are talking shit.
I can see the parrot of shame lmao, you're not slick.
Of course that thread is full of parrots, what else did you expect?