Hi, Grishka,
Up above, Mark Cohen says he learned the song as a round in about 1986. Here's a recording of the song as a round:
http://thebirdsings.com/songs/alfred.html

I found a March 1967 magazine that says that Peter Tork sang two songs written by his brother Nick Tork:

...I must tell you about one particularly funny night when I dropped into the Playhouse Cafe while Peter was working alone.

"And now, ladies and gentlemen," I heard him saying, "I'm going to sing you a couple of original tunes. You may not know it, but I have a little brother who is a great songwriter. As a matter of fact, he's working on a complete musical score for the theatre right now. His name is Nick Tork, and you'll be hearing a lot about him one day." Then he launched into one of the most ridiculous and funny tunes I have ever heard. All I can remember about it is its title—"Under The Undertaker Gah Goo-Gah." When the audience "revived", Peter launched into another of his brother's ditties—all about an alligator named Albert (sic), who got flushed down the toilet! Well, some of the Village cats may have sneered at that, but the audience ate it up—and Peter got even more "clinks" and "swishhhs" in his banjo!

Monkee Music by Andrew Hickey says "Alvin" is a 24-second a cappella recording of a song written by Peter Tork's brother Nicholas Thorkelson, about missing a pet alligator who's been flushed down the toilet. Lead singer on the recording was Peter Tork (and I hear only one voice on the recording.

-Joe-