I think I've found the one song utterly immune to parody, unless you count unconscious self-parody. That would be the traditional Thanksgiving favorite, "We Gather Together...." YouTube provides no Muppet version, no Klezmer, no Tom Waite, none illustrated by Munch's Scream. Just straight-up old-fashioned American kitsch.
Except not quite American. Wiki explains that it's originally Dutch, apparently first a folk tune, refashioned into a hymn in 1597 to celebrate a military victory over the Spanish. The author is one Adrianus Valerius, poet and toll collector and author of an anthology, Nederlandtsche gedenck-clanck. The familiar American text dates only to 1894. The English internal rhymes ("chastens and hastens") mirror the original Dutch ("boven am loven"). There are other English renderings; one begins "We gather to worship Jehovah, the righteous,/Who verily sitteth in Jugdment severe."
Wiki reports that the hymn evidently gained popularity during World War II, when its talk of "the wicked oppressing" was taken to refer to thr Germans and the Japanese. The original circumstances were less triumphal. The Netherlandish commander whom Wiki describes as "the ever prudent [Prince] Maurice" proved imprudent on the occasion of the original battle and the war extended for another 51 years.
Lacking a parody rendition, here's the least parodic rendition I can find: link.
Meanwhile, if you need some real genre-bending, go listen to what Uri Caine does with Mahler's Songs of a Wayfarer.