I don't know for sure. Peter would have been the right age to be the son of a Loyalist. Charlotte (nee TWADDLE), his wife was b. ca 1784, according to her gravestone in St. Thomas Ang. Church Cem., Black River,NB. Their two sons I know of are John and Benjamin.
I lean more to Peter being the s/o Benjamin APPLEBY. In Ben.'s probate record ( Book - Early NB Probate Records by R. Wallace Hale) he died in the Parish of Wickham,Queens Co.,NB probated 1827. "Three sons, Peter, John and Benjamin, each 5 shillings. Daughter Mary WANAMACKER, ½ my household furniture. Son James residue of my estate.
Peter APPLEBY,(prob. Loyalist) Parish of Hampton,Kings Co.NB,Intestate. Admin granted 5 Nov 1829.Eldest son John.Bondsmen Salyer MORRELL and Henry WANAMACKER.