It is quite obvious from the Mecklenburg Co, VA citations that I shared, Peter Burton, son of Abraham Burton and Ann Neal, is the Peter Burton, who died and left a will in 1796 in Mecklenburg Co, VA, where he clearly names his parents Abraham and Ann Burton. Whoever came up with the conclusion you question, clearly was in error. If you look at the Rootsweb.com Familytrees for Peter Burton, died in 1834 in Haywood Co, TN, most researchers have his parents as Peter Burton and Martha Cousins. The same is true in Ancestry.com Public Member Trees. I think I may have seen one researcher amongst them all place this Peter Burton (d. 1834) as son of Abraham and Ann Burton. As to who are the parents of this Peter Burton, of Stokes Co, NC and Haywood Co, TN, I cannot say for sure. There were several Peter Burtons in Amelia Co, VA and in Mecklenburg Co, VA, and I still have questions. However, I have TENTATIVELY decided to assign this Peter Burton (d. 1834) as son of Peter Burton, who appears to have been married to Elizabeth Cocke and who died about 1765 in Amelia Co, VA. But this is really quite speculative, with many question marks.
This Peter Burton (d. 1834) first appears on the scene in Stokes Co, NC in 1794, with his marriage to Rebecca Guinn. In 1817, this Peter Burton sold his land in Stokes Co, NC and then purchased land in Williamson Co, TN . By 1834, when he signed his will, he had migrated farther west to Haywood Co, TN. I have found nothing to indicate the actual parents and birthplace of this Peter Burton. But for sure, he ain't the son of Abraham Burton and Ann Neal.