Peter Moore and Mary Cottier Moore are the parents of Jane Moore, who married Rev. John Cottier.
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Marriage:
Name John Cottier
Spouse's Name Jane Moore
Event Date 09 Sep 1810
Event Place Saint George,Douglas,Isle Of Man,England

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Their son, Claudius Thomas Cottier, who emigrated from the IOM to New Orleans, LA in 1851, is my g-g-grandfather.

According to the LDS Family Search link below, there is Pedigree Resource File which has an MI for Peter Moore and Mary Cottier Moore, who died in 1831/01/14, at age 70. The MI lists their place of burial at Kirk Braddan, Old Yard, South.
familysearch.org/ark:/61903/2:2:SPPL-GFM

The Pedigree Resource File on Family Search gives you the names of his children and the names of his parents, but there are no parents listed for his wife, Mary Cottier.

In looking at the marriage record image for Peter Moore and Mary Cottier, the witnesses were an Edw. Corkhill and John Cottier.
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Marriage record image link:
familysearch.org/ark:/61903/3:1:S3HT-6PT4-4S?mode=g&i=45&cc=1784428

Name Peter Moore
Event Type Marriage
Event Date 19 Nov 1780
Event Place Maughold, Isle of Man
Gender Male
Spouse's Name Mary Cottier
Spouse's Gender Female

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Here is the 1760 Maughold, IOM christening record for a Mary Cottier, with father, John Cottier listed. I'm guessing, but really don't know for sure, that is the Mary Cottier who married Peter Moore and her father, John Cottier, was one of the witnesses to her marriage.
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Baptism record image link:
familysearch.org/ark:/61903/3:1:3QS7-89HK-KHR
Isle of Man Parish Registers
Name Mary Cottier
Event Type Baptism
Event Date 14 Sep 1760
Event Place Maughold, Isle of Man
Gender Female
Father's Name John Cottier

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I'd be grateful for any help with this Peter & Mary Cottier Moore family.