Once you find an ancestor listed in the Social Security Death Index, you do not need to stop there. You can request of copy of your ancestor's application for a social security card if you wish. The form is called the SS-5 and it often contains parents names as well as other detailed personal information. Here is a list of the info on the form:
Full name at birth (including maiden name)
Present mailing address
Age at last birthday
Date of birth
Place of birth (City, county, state)
Father's full name "regardless of whether living or dead"
Mother's full name, including maiden name, "regardless of whether living or dead"
Sex and race
Ever applied for SS number/Railroad Retirement before? Yes/No
Current employer's name and address
If you are using Ancestry.com to search the SSDI, once you find the record you can request the original application from an option to the left of the record showing on the screen. Ancestry will even generate the appropriate letter that you can then sign and send. There is a cost to get the record, but it is only about $25-$30.