The Draft Registration Cards now online are over 65 yrs old, and the Government figures all are dead by now so no ID Theft possible and Privacy Laws are not broken by releasing them. The time frame of your dad's won't be released till after 2030. If your dad is still alive, only he has access to any Military records anyway. After his death then only his next of kin as defined by NARA can have access.
The Selective Service Agency is not the "Military" and should have other storage not archived at NARA, like Vietnam Era Draft Registrations. That is why the WWI and WWII Old Mens And some WWII "Young" Mens Registration Cards are available to scan and upload.
There was a fire in 1973 at the NARA, but only 80% of Army/Air Corp, Air Force were affected up to early 1960's. The NARA website has all the information about it and how to gather other records for information. Navy, Marine, Coast Guard were not stored at NARA in 1973 in St. Louis, so all records are available. www.archives.gov