If you think Thomas Gaskins was his father, find Thomas's probate records. All of his children should be listed there.
You'll find the county records on microfilm at the Virginia Library, www.lva.virginia.gov/public/local/results_all.asp?CountyID=V...
If Thomas is not Edward's father, then you'll have to look for Edward's baptism record. You can contact the following and ask if they church records: Northumberland County Historical Society, http://northumberlandvahistory.org/
. They do have Gaskin in their surname index. Northern Neck of Virginia Historical Society, www.nnvhs.org/
I'd look for Edward's marriage document. His parents names might be on it. Also look for his newspaper marriage announcement. The Library of congress, Chronicling America, has old, free, digitized newspapers.
Edward W. Gaskins
Born: abt. 1817, Lancaster
Spouse: Eliza Gaskins
Died: 4 Sept 1871, Lancaster, VA
Source: Va Deaths & Burials Index 1853-1917
I'd contact the above two societies and ask about his newspaper death notice. His parents names might have been in it too. Successful Searching!