I have some ideas and some data. Perhaps not enough to be conclusive, but it does make a good circumstantial case. I do not think that this is Thomas' father. Here is part of a document based on Maryland records.
Abraham4 Cowardin (Thomas3, Thomas2, Peter1) was born after 1716 in Kent County and married Mary
Joce on 2 November 1738. 1738 was a great year for young Abraham. In addition to getting married that year, he also inherited the 300 acre plantation Mount Herron Resurveyed from his kinsman William Thomas, and inherited the 300 acre plantation Poplar Farm from his cousin Sarah Evans. According to Kent County Debt Books, he appears to have owned Poplar Farm at least until 1760. At June Court in 1744 he was fined 5 pounds for selling liquor. The next year at June Court he was fined 10 shillings for assaulting Joseph Thomas (his cousin). Abraham was the father of: John, born 1 December 1739; Martha, born 2 July 1741; Pasco, born 25 September 1744; Sarah, born 29 June 1747, and Abraham, born 2 October 1751.
Mary Joce was the Daughter of Nicholas and Jane Blackiston Joce. The Blackistons and Joces were some of the earliest settlers in the area. They were of prominent standing by the time Abraham married into the family. There is no record of death for Abraham or any of his family. They probably moved.
Records for Abraham and his family disappeared from Maryland in the early 1750's. In the late 1750's, Abraham appears in South Carolina. I think that these are the same person.
What is the source of the Parkins story? Do we know what year this happened?