Trendy Parisian quickly making request for company (6)
I believe the answer is:
invite
'request for company' is the definition.
'trendy parisian quickly' is the wordplay.
'trendy' becomes 'in'.
'parisian quickly' becomes 'vite' ('quickly' in French).
'in'+'vite'='INVITE'
'making' is the link.
(Other definitions for invite that I've seen before include "Encourage, ask" , "Ask (someone to visit)" , "Ungenerous" , "Ask to call" , "Ask to one's home in friendly way" .)