That got me interested in your airport. Based on the information in the FAA database when you fill out the FAA Form 7480-1 "FAA Notice for Construction, Alteration and Deactivation of Airports" you have in #10 "Current Use" and the options are : Private, Public, or Private use of Public Lands. I gather your parents checked Public ( I checked Private). Then under #11 "Ownership" the options are : Private, Public, Military. I would think they checked Private ( So did I). So my best guess based on the database is that you have a private airport with public use, like where I have my airplane right now 3F9 in Texas, but it wouldn't be the first time I'm wrong, lol.
If not you might want to check and correct the database. I had to correct it for mine and it still has some mistakes. It could also just be a technical term thing and we're both talking about the same thing just using different terms.