Concedes the tailless swallow looks very odd (6)
I believe the answer is:
allows
'concedes' is the definition.
('allow' can be a synonym of 'concede')
I don't understand the rest of the clue.
Can you help me to learn more?
(Other definitions for allows that I've seen before include "Sallow (anag.)" , "Grants leave" , "Permits, lets" , "Permits to do something" , "Tolerates" .)