I have some information on Travis' descendants, but have not yet been able to find his parents.
Travis was in Guilford Co NC on the 1820 census, and in Chatham Co NC on the 1800 census. (I haven't found him on the 1810 census.)
I'm pretty sure all the NC Rigsbys were fairly closely related, and that they were all originally from VA. There were three Rigsby immigrants to VA before 1700: John, James, and Thomas. Unfortunately, I haven't been able to find any record of their marriages or children.
My current guess is that Travis was a brother of Drury Rigsby who served in the Revolutionary War, Luke Rigsby of Chatham Co, and Jesse Rigsby of Orange Co. I have no proof of this, though.
From what I have found, your line changed from Rigsby to Rigsbee with Marshall Benton Rigsbee. His father was Rigsby on the censuses up to 1870, and then in 1880 is Rigsbee. I am never sure if the Rigsbys themselves changed the spelling, or if it's just a census enumerator spelling phonetically.