Aspirin habits developed, quietly supporting two parties (14)
I believe the answer is:
bipartisanship
'supporting two parties' is the definition.
(supporting or working with two political parties)
'aspirin habits developed quietly' is the wordplay.
'developed' is an anagram indicator.
'quietly' becomes 'p' (p for piano and quiet).
'aspirin'+'habits'='aspirinhabits'
'aspirinhabits' is an anagram of 'bipartisanshi'.
'bipartisanshi'+'p'='BIPARTISANSHIP'
(Other definitions for bipartisanship that I've seen before include "Inclusive approach" , "adherence to two groups" .)