I would say that it's not a parody. A parody is generally some sort of a piss-take, even if fairly reverential, of the original. Further, "Hard Times" goes back almost as far. I'm guessing that the fact that it's a "parody" (by your definition) escapes most people who sing it today. Further again, it's a song with serious intent, not quite what you'd expect of a parody. My friend Martin Cole and I recorded this 20 years ago. He sang and played the guitar and I played a diatonic harmonica in G (which I had to retune to get a low note!). It didn't make it on to our CD but it could have done. It never occurred to me that it was some sort of parody. I'd also remind you that simply recycling a tune with different words doesn't necessarily add up to a parody. Woody Guthrie reused tunes for his songs all the time. If a song carries an important message, who cares if it's using an old tune?