Serious crime of beheaded parish priest (5)
I believe the answer is:
arson
'serious crime' is the definition.
(crime of starting a fire)
'beheaded parish priest' is the wordplay.
'beheaded' means to remove the first letter (the head of the word is removed).
'parish priest' becomes 'parson'.
'parson' with its first letter taken off is 'ARSON'.
'of' acts as a link.
(Other definitions for arson that I've seen before include "major crime" , "Incendiarism" , "Fire-raiser's crime" , "Felonious burning of property" , "Act of a pyromaniac" .)