Any gift of worthless or negative value; i.e., like a soon-to-die sick bird or an already dead parrot that's been given as a gift. Metaphorically speaking, a kind of backhanded compliment or ironic flattery meant to mock or piss someone off (as one would be pissed off to receive a sick bird as a gift).
"What do you think of me going for that new job HR just posted?"
"I think your monumental intellect could only get in the way of your minimal experience."
"Ah, why the sick bird, dick?"