The marriage record for his marriage to Matilda Coldwell shows his parents as Robert and Eliza Blanton.
There is a marriage of Robert F. Blanton to Eliza Irvine (probably marriage license) and Eliza Jeonie (probably bond) in 1892. You will probably need to get the original to find out if one of them is her maiden name.
Note that the marriage date is consistent with John Lukey's birth date of 17 Jan 1893, as listed on his WWI draft registration.
You will also want to get his SS# from the Social Security Death Index, so you can order his SS application. This is the best source for parents' names, although it costs a bit. You can order it here:
https://secure.ssa.gov/apps9/eFOIA-FEWeb/internet/main.jspThe 1900 census record in Edmonson that many researchers cite is probably a different John L. It is pretty far from Harlan and over the mountains. The names do not match those in the 1910 Knox census. There were a lot of Blantons there from an entirely different group (Tidewater Blantons). There would be no reason for the children to be with their grandparents, since Robert was alive until about 1908. The family was probably simply missed.
All of the databases mentioned can be found at familysearch.org