The most common and most likely answer to this crossword clue is the 1 plus 3 letter word A BAS
A
B
A
S
More clues leading to the result A BAS
Buddy explains!

The straight part
We know that "down in Paris" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case down in paris = a bas.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.