This song was written by Christopher Cerf: writer, lyricist/composer, vocalist, parodist/satirist, and all-around hero to me. He wrote music and/or lyrics for some other songs that I've already featured, such as "Dance Myself to Sleep," "I Wish I Had a Friend," and "Bert's Blanket."
Here are some other Sesame Street songs that I love, not involving Bert & Ernie, from the pen of Chris Cerf:
You're Alive (featuring Cerf's Muppet doppelganger, Little Chrissy)
Long Time, No See
Finally, Telephone Rock, which is significantly dated, but I get a vicarious thrill from the ending.
Needless to say, I really love his music. His parodies of specific songs are always spot-on; I would go so far as to say that some of them even improve upon the originals, although that's a matter of opinion. Most of his songs have an element of parody, or at least an homage to a genre of music, but many of his original songs are simultaneously hilarious and heartbreaking.