Woman showing a mother and father around centre of Doonbeg (6)
I believe the answer is:
amanda
'woman' is the definition.
'a mother and father around centre of doonbeg' is the wordplay.
'mother' becomes 'ma' (Ma is another word for mother or mum).
'and' means one lot of letters go next to another.
'father' becomes 'da' (da can mean dad).
'around' indicates putting letters inside.
'centre of' indicates the central letters.
The middle letter of 'doonbeg' is 'n'.
'ma'+'da'='mada'
'mada' going around 'n' is 'manda'.
'a'+'manda'='AMANDA'
'showing' acts as a link.
(Other definitions for amanda that I've seen before include "Girl's name meaning lovable" , "Girl" , "F. name meaning lovable" .)