You said in your message that you have your dad's social security number; have to wait several years but with this number you can get a copy of his original social security application form that he himself either filled out or directly gave the information to someone to fill out the form....BUT they will not give it out with the parents' names until the person is over 100 years old. Therefore, you'd have to wait until Nov of 2022....sorry, but they recently changed the parameters. These forms have the father's name & the mother's maiden name.
But these forms are oftentimes used as proof of parentage if no birth cert. exists.. and oftentimes better than a death cert. as the person himself gives the information rather than an informant, like a child, after the person's death.
To get the form: ssa.gov foia - then go to "Make a FOIA Request" - then to "Frequently Requested Records" and then choose Form SSA - 711. Get the photo copy, NOT the abstract.... if the written form is hard to read, you'd have more interest in figuring out what the record says rather than someone who is just getting thru a pile of entries.
I found a few records for Harry that you may not have found?
1) Was he in the military? If so, on familysearch, there is a WWII enlistment record for a Harry E Simon, b. 1922 in SC, with enlistment date of 04 Feb 1942. If this is his, this means that he has a military file that you could request copies of. In many cases, this file will have his parents' names or a birth cert. or an attestation for his birth information.
2) I find two social security records for Harry Simon with a death date of Dec 1970 but both with a birth date of 11 Nov 1924. The social security number with these records is: 249-16-9663. If this is a match to what you have, you may need to use this birth date for your FOIA request. ...which would make it 2024...sorry again!