Parish priest's heading off — it's a crime (5)
I believe the answer is:
arson
'a crime' is the definition.
(crime of illegally starting a fire)
'parish priest's heading off' is the wordplay.
'parish priest' becomes 'parson'.
'heading off' means to remove the first letter.
'parson' with its first letter removed is 'ARSON'.
'it's' acts as a link.
(Other definitions for arson that I've seen before include "Pyromaniac crime" , "Malicious burning to destroy property" , "Crime involving fire" , "matchplay, say?" , "Act of a pyromaniac" .)