There is a Declaration of Intention for naturalization from 1921 for Cecil Theodore Martin. It says he was born Georgetown, British Guiana on 21 Aug. 1885. Current address 323 West 37th St., New York City, New York. Occupation was compositor. Complexion - brown, hair - black, eyes - brown, 138 pounds. It says his last foreign residence was Cuba & he sailed from Havana on the vessel Morro Castle to New York, arriving 20 Nov. 1919. It says his wife Bessie was born in the US and living with him.
A Petition for Naturalization dated 21 Dec. 1926 gives the same date & place of birth. His residence was 210 W. 138th St., and his occupation was realtor (which matches the census records and the draft registration card). This says he sailed from Havana to New York on the Morro Castle from 8 Mar. to 12 Mar. 1917. His wife Bessie was born Norfolk, Virginia in 1895 and was living with him. He had no children. It appears he was naturalized on 2 May 1927.
It looks like his year of birth has fluctuated a bit, but the 1885 date seems to match this result on the Social Security death index -
https://familysearch.org/pal:/MM9.1.1/VMKF-82L If this is him, you could order his original Social Security application and it would likely include his parents' names. More information on how to do this is here -
http://genealogy.about.com/od/online_records/a/ss5_request.htm If you know where he died, then ordering his death certificate would be another option. If it was New York, you can order recent death certificates if you can prove you are a direct descendant. Please note that New York City keeps its vital records separately from New York state, so be sure to order from the right place.