Father beginning to tire of island's dialect (6)
I believe the answer is:
patois
'dialect' is the definition.
(patois is a kind of dialect)
'father beginning to tire of island's' is the wordplay.
'father' becomes 'pa' (pa is a kind of father).
'beginning to' says to take the initial letters.
'island's' becomes 'is'.
The initial letters of 'tire of' is 'to'.
'pa'+'to'+'is'='PATOIS'
(Other definitions for patois that I've seen before include "Dialect or jargon" , "Spoken regional dialect" , "A regional dialect of a language" , "Form of language" , "Language of group" .)