Strictly speaking, I don't think you can call this a parody because it uses a different tune than THE SWEETHEART OF SIGMA CHI. Only the title harks back to the earlier song.

You can hear this song at The Internet Archive:


SHE'S THE SWEETHEART OF SIX OTHER GUYS
Words and music by Howard Johnson and Robert King.
New York: Triangle Music Pub. Co., Inc., ©1928.
As sung by Fred Hall and His Sugar Babies

Hail, hail, the gang's all here!
What the heck do we care?
What the heck do we care?
One, two, three, four five,
Six other guys are worrying me.

Oh, my, oh what a sweetheart I've got!
But she should be shot, I claim.
Oh my, she says she likes me a lot,
And maybe she's not to blame.
I can't really love her because
Of just a few things that she does:

On Monday she takes a walk with Johnny.
On Tuesday she has a talk with Jim.
On Wednesday she likes to dance with Tommy.
On Thursday she takes a chance with Tim.
On Friday she goes to dine with Billy.
On Saturday it's Percy that buys.
Sunday's my only time with that bozo of mine,
'Cause she's the sweetheart of six other guys.


[I ran across this song because Billy Jones and Ernest Hare are supposed to have recorded it, but I can't find a recording of theirs online.]