Are songs in waltz tempo more open to parody?

Probably because it is “bouncier” than other tempos.

The time signature for The Battle Hymn of the Republic is 4/4. So you have disproved your own hypothesis.

I can’t imagine why there would be any relationship between time signature and openness to parody. So, absent theory and absent any evidence suggesting such a relationship, I’d say the answer to this question is an unqualified NO!

@marinelife ,Thanks for the correction. My music talent is definitely deficient. I still think it would be of interest to see if there are certain characteristics of music, apart from lyrics, that make them more likely to being parodied. And I am not giving up so fast on my waltz hypothesis, even if there are exceptions.

From off the top of my head, here are songs that I heard parodied as a youngster. I looked up the signatures.

Turkey in the Straw – 2/4
My Country Tis of Thee – ¾
Dance of the Hours (hello, muddah) – 2/4
Clementine – ¾
Stars and Stripes Forever – 4/4
Bridal Chorus (here comes the bride) 2/4
Twinkle Twinkle Little Star – 2/4

My survey turns up a predominance of 2/4, lending credence to @zenvelo ‘s suggestion.

There’s a “waltz” in 5/4 time in Tchaikovsky’s 5th Symphony.

Songs in ¾ time that come to mind as being often parodied in old cartoons are “Ach du Lieber Augustin; “Over the Waves” (played on a trombone when somebody is drunk or seasick); “Blue Danube” waltz by Strauss, which lends itself nicely to certain sound effects; etc. I think a lot of children’s songs are in ¾.

And don’t forget “Happy Birthday to You” which is also in ¾ time. Do those 2 old ladies still get royalties when it’s played in movies?

As a final comment, consider the trimeter structure of limericks.