These examples work in any POSIX shell and require no external programs.
This stores the Part*.mp3 files at the same level as the Project directory (provided there are no files/directories already there w/ the same names):
(cd Project && for i in Part*/audio.mp3; do echo mv "$i" ../"${i%/*}".mp3; done)
This keeps the Part*.mp3 files in the Project directory (w/ the same caveat about pre-existing names as the other):
for i in Project/Part*/audio.mp3; do echo mv "$i" ./"${i%/*}".mp3; done
These solutions use the shell's pattern matching parameter expansion to produce the new filename.
${parameter%%wordparameter%word} Remove LargestSmallest Suffix Pattern. The word is expanded
to produce a pattern. The parameter expansion then
results in parameter, with the largestsmallest portion of the
the suffix matched by the pattern deleted.