For Christmas 1965, I got the Mary Poppins soundtrack album and a comedy LP recorded by Fred Gwynne, Yvonne DeCarlo, et al. I still can sing “Supercalifragilisticexpialidocous” and the theme from The Munsters. (Yes, there are lyrics.)
I don’t understand this thread at all. Why does “that AREN’T parodies” matter? Is the intention just “what songs do you know the words to that you believe are known by the fewest other humans on the planet?”? (Is the disclaimer intended to mean “other than parody songs you yourself wrote and thus literally no one else could possibly know?”)
In any case, there’s a difference here between things that are well recorded by history but very few people know off the top of their head (ie, second verse of the Star Spangled Banner) and things that will someday soon, when a few more people die, entirely vanish from the human record, and you’re one of those people… or at least, as far as you know that might be possible.
In the latter category, my 80-year-old-father sang to me a while ago the camp song for Camp Big Paw, a summer camp he attended in the 1950s somewhere near Chicago. When he dies I may be the only human being left who can sing the Camp Big Paw theme song. Or maybe it thrived for decades, or even still exists, and thousands of Chicago-area kids sing it all the time. Beats me.
One somewhat relevant story… I grew up singing the 1950s-era comedy songs of British duo Flanders and Swann. They’re not obscure, per se, but they’re certainly not well known, in the USA, among people born after 1970. One day in college (1990s) we were playing a “come up with songs that fit a theme” game of some sort, and I started singing Mud, Mud, Glorious Mud, and someone else started singing along, and I nearly fell over in shock. (We dated for several years, although I can’t remember which came first.)
60 years later I can still sing the first verse of “Praise God from whom all blessings flow”. It was my favorite song, because when it started up it signaled church service was over.
…Love, strange love a star-woman teaches…
I can sing Weird Al’s “Bob.”
Ah, a kindred spirit! 
Tenacious D has covered the latter.
I learned “The Battle of Trenton” by Mike Agranoff several decades ago on a hiking trip and can still recite it today. Thinking about this, I looked it up and found his performance of it on Youtube, along with a few other favorites. I need to brush up on “The Ballad of Jake and Ten-Ton Molly”.
Dave Brubeck’s wife wrote lyrics to many of his tunes… i remember
the first 2 lines of “The Duke”, but was vomiting too much to
read any more.
Yes - they are that bad.
Key did write four verses for the Star-Spangled Banner. But many others have added verses, and the third verse is often deliberately left out because it has the word “slaves” in it. Nobody’s absolutely sure but it probably refers to slaves who joined the English who promised them freedom.
The song has a long, weird, and complex history I just read about in O Say Can You Hear?: A Cultural Biography of "The Star-Spangled Banner. Fascinating book mostly. (Some chapters are so specialized about music theory I got lost.)