The most common and most likely answer to this crossword clue is the 4 letter word RSPB
I've seen this crossword clue in The Guardian
R
S
P
B
Buddy explains!
The straight part
We know that "They rescue sick parrots, bandaging heads" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they rescue sick parrots, bandaging heads = rspb.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.