The names were not just parodies but also meant as promotional tools. Warner Brothers, along with the other major Hollywood studios like MGM and Paramount, owned a music publishing company in order to secure music copyrights for use in their film and cartoons. When the Looney Tunes and Merrie Melodies series began in the early 1930s, the cartoons were often used to showcase a song in Warner’s extensive music library. The idea would be the audience members would like the song enough to go buy the sheet music (which still comprised a large share of the money earned in the music industry) and add some more coinage to the corporate coffers.
For years, I thought songs like “We’re in the Money!” and “So This Is Gay Par-ee!!” were written exclusively for the cartoons. I was surprised to hear them in actual old movies with live actors. 
Same thing when I first discovered Harry Warren: he wrote most of the songs for WB musicals, and I had heard most if them on Merrie Melodies.
Or “The Daughter of Rosie O’Grady” (which Bugs sings at one point)
“…A regular old-fashioned goil!” 
Something else I noticed in the above classic video explains a line in a Jim Croce song that I assume was something of a stereotype in 1943, which never would have occurred to me:
The “Mean Old Queen’s” coat of arms features a pair of crossed razors that aren’t in real good shape.
I would love to see sheet music for “Hillbilly Hare”. Someone has attempted it:
You should have seen me when I saw Wagner’s Ring Cycle and the Giants’ leitmotif came up the first time.
“Gossamer!”
My late brother was a classical music fanatic. Watching “Kill the wabbit, kill the wabbit, kill the wabbit!” he was laughing harder than anyone else in the cinema.
Classic! 
Also classic! 
“You know, it is so sad. All your knowledge of high culture comes from Bugs Bunny cartoons.” (Elaine Benes to Jerry Seinfeld)
The joke that mystified me as a kid was in “High Diving Hare”. Yosemite Sam is pounding on a door, saying “Open the door!” and then he follows it by saying “You notice I didn’t say Richard?”. It was years later that I learned that there was a popular song “Open the Door, Richard!”.
My kids learned about opera the same way I did.
From a cross dressing rabbit.
It was a favorite song when I was a kid in the 1950s. This is the version I know.
“Open the Door, Richard” was the 1950s version of “Dave’s not here!” (if people still get that one).
Some time after I figured out the reference, I was watching that cartoon with my dad in the room and (showing off my know-it-all-ness) I told him that it was a reference to a song. He responded that he was familiar with the song, but he was probably thinking “gee thanks, Captain Obvious”.
I did the same thing when I learned where the term “riding shotgun” originated. I think when I explained that it was a reference to a guard who sat next to a stagecoach driver holding a shotgun, his response was something like “Really, you didn’t know that before?” I mean, he grew up watching Westerns when he was a kid. By the time I was born the age of the Western was more or less over.
My dad was clearly a child of the 1920s. He used to talk about riding shotgun on a beer truck, and it was years before I got the reference.