This isn't so much a parody as one version of a whole family of related folk songs, some of which are given in this thread: Lyr Add: Mister Boll Weevil.
Oh, the boll weevil an' the little black bug come from Mexico, they say, Come all the way to Texas, just a-lookin' for a place to stay, Just a-lookin' for a home, just a-lookin' for a home.
The first time I see the boll weevil, he was settin' on the square. The next time I see the boll weevil, got all his fam'ly there, Just a-lookin' for a home, just a-lookin' for a home.
The farmer say to the weevil, "What makes your head so red?" The weevil say to the farmer, "It's a wonder I ain't dead, Just a-lookin' for a home, just a-lookin' for a home."
The captain say to the missus, "What do you think of that? The boll weevil done made a nest in my best Sunday hat. Why not have a home? Why not have a home?"
The boll weevil got half the cotton. The merchant got the rest. Didn't leave the farmer's wife but one ol' cotton dress. It's full o' holes. It's full o' holes.
If anyone should ask you who was it made this song, Just say that Homer and Jethro have done been here and gone, Just a-lookin' for a home, just a-lookin' for a home.
[Originally recorded by Homer & Jethro as a 78-rpm record, (a) DON'T LET YOUR SWEET LOVE DIE, b/w (b) BOLL WEEVIL, King 583, 1947. It is also on their album "The Best of Homer & Jethro: Hall of Fame 2001," Federal Records 501, 2002.]