Here's the version that's in the DT, where it's credited to Fred Allen. As you can see, it has two additional verses -- each of which, for some reason, seem to be missing a final line.

Anybody care to guess what the missing words are, or why they're not there? My guess is that the last line in the second verse probably ends in "leather."

Be kind to your web-footed friends,
For the duck may be somebody's mother:
Be kind to the denizens of the swamp.
Where the weather is cool and damp

Be kind to your old umbrella,
For some day it may be under the weather,
Be kind to your old pair of shoes.

Be kind to your fur-bearing friends,
For a skunk may be somebody's brother-;
Be kind to your friends with the stripe.