It was different for my parents though. According to the above mentioned site, my dad got his SSN in 1971. This was his first job (outside of working for his dad), when he went to work for a gas station at age 16. This means that he went down and personally filled out his own social security application. He would have personally recorded his name, birthday, address, parents names, etc. This is the same process that his parents and grandparents went through to get their social security numbers.
The Social Security Act was signed in 1935, so your ancestors in the workforce at this time should have filled out an application form. Some folks, like farmers or homemakers will not necessarily have done this. Search the Social Security Death Index for your ancestors. If they appear in this database, you should be able to request their application form. Even if they don't appear there, you might still find a form for them. The forms cost $27 when you know the SSN and can be requested here: Social Security Online.
Case Study: Here's my Great-Grandfather Vary "Mack" Huyler's social security application. In 1937 Mack was AWOL from his family, having left to "find work" and neglecting to return for 14 years. Before requesting this document, none of his descendants knew the names of his parents. From requesting this document, I was able to begin tracing the Huyler line back to the 1600s. This application provides a wealth of information about him: address, job, birth day, parents names, etc, that have really helped in my research.