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The above birth might be the one - but Shirley is right, you need her marriage cert to your father:

Name: Norman Peacock
Spouse Muriel Thornton
Date of Registration: Jul-Aug-Sep 1943
Registration District: Skipton
Inferred County: Yorkshire West Riding
Volume Number: 9a
Page Number: 115

you have the marriage date but do you have the actual cert??
Name: Suzanne Peacock
Spouse Surname: Fairhurst
Date of Registration: Oct-Nov-Dec 1964
Registration District: Staincliffe
Inferred County: Yorkshire West Riding
Volume Number: 2d
Page Number: 633
Find Spouse:Robert G Fairhurst

first birth that comes up for her name

Name: Suzanne Peacock
Mother's Maiden Surname: Thornton
Date of Registration: Jul-Aug-Sep 1944
Registration District: York
Inferred County: Yorkshire
Volume Number: 9c
Page Number: 1684

Have you got the marriage cert to your dad that should give her fathers name and her age at marriage

Hi, I am trying to trace my mother, maiden name SUZANNE PEACOCK.(dob: c1940?) I have a feeling my mother's parents lived in Skipton, North Yorkshire, but not 100% sure. (i don't know their names. I don't even know if my mother had any siblings.)
She married my father ROBERT GUY FAIRHURST on 3/10/1964 (in the district of Staincliffe), and was known to have been living in Colne, Lancashire at that time. (I was born in 1966)
After they divorced, I was brought up by my Grandmother Sarah Eleanor Fairhurst (Nellie) and my father in Kelbrook. (my mother was never discussed)
I heard from someone that my mother had remarried, possibly had more children and was living in Skipton, also heard that her new husband had sadly died around Christmas one year. I don't know if she remarried after that.
Rumour has it that she moved from Skipton in the early 1990's but I don't know where to.
I've not had any contact with my mother in over 40 years, so if anyone knows anything about her, or any of my half siblings, please get in touch.

Thank you in advance.

(Unfortunately, i can not ask my father as he passed away in 1983)