I backed European parliament? That makes a great story (5)
I believe the answer is:
iliad
'a great story' is the definition.
'i backed european parliament?' is the wordplay.
'backed' shows that the letters should be reversed in order.
'european parliament?' becomes 'dail' (Dail Eireann, part of Irish parliament).
'dail' back-to-front is 'liad'.
'i'+'liad'='ILIAD'
'that makes' acts as a link.
(Other definitions for iliad that I've seen before include "Poem by 4 across" , "epic work" , "heroic work" , "set of books" , "Homer's work" .)