The most common and most likely answer to this crossword clue is the 9 letter word NOSEBANDS
I've seen this crossword clue in The Guardian
N
O
S
E
B
A
N
D
S
More clues leading to the result NOSEBANDS
Buddy explains!
The straight part
We know that "bridle parts" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bridle parts = nosebands.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.
Crossword Answer definitions
NOSEBANDS noun- The part of a bridle or halter that goes over the nose of an animal, particularly a horse.