Francis and Mary are the most likely parents for Jane, this is the only baptism found for "Buxton" Manchester in 1815. Francis was a Fustian cutter - a cutter of corduroy, which would link with Edward who was a tailor
Sources
1815 Baptism Jane Buxton
1858 Burial records
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DNA Connections
It may be possible to confirm family relationships with Mary by comparing test results with other carriers of her mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Mary: