I don't know if PP&M recorded it, but the "other" Unicorn Song is by Margie Adam, and it's great.

When I was very small/ My best friend was a Unicorn;/ The others smiled at me and called me - crazy;/ But I was not upset/ By knowing I did not conform/ I always knew their thinking must be - hazy/

(Chorus) The unicorn and I would while away the hours/ dancing prancing and romancing in the wild flowers and we'd sing/ seeing is believing - in the things you see/ Loving is believing in the ones you love.

She was pianist and singer/songwriter around in the 70's; I have no idea what she's up to now.