And, apparently, a kid's parody was written right into this commercial:

Subject: RE: Jingles you remember.
From: Mark Cohen - PM
Date: 28 Jun 04 - 04:01 AM

...

And remember the little kid walking down the street singing,

Oh, I'm glad I'm NOT an Oscar Meyer wiener,
That is what I never want to be
'Cause if I were an Oscar Meyer wiener
There would soon be nothing...left...of...me...

At which point he ran into a crowd of kids with one girl who gave him a dirty look, and he sheepishly joined the crowd of kids singing,

Oh, I wish I were an Oscar Meyer wiener
That is what I'd truly love to be
'Cause if I were an Oscar Meyer wiener
Everyone would be in love with me

...