Scoring parts of their cartoons (13)
I believe the answer is:
orchestration
'scoring' is the definition.
(I've seen this before)
'parts of their cartoons' is the wordplay.
'parts of' is an anagram indicator.
'their'+'cartoons'='theircartoons'
'theircartoons' with letters rearranged gives 'ORCHESTRATION'.
(Other definitions for orchestration that I've seen before include "Musical arrangement" , "Scoring for strings, brass, woodwind and percussion" , "Composer's skill" , "Arrangement of string, brass, woodwind and percussion parts" , "Arrangement for instruments - short reaction (anag)" .)