How about
Allan Sherman - Hello Muddah Hello Faddah (1963)
Pepino the Italian Mouse - Lou Monte - 1963
Martian Hop - The Ran-Dells - 1963
Harry the Hairy Ape - Ray Stevens - 1963
Puff the Magic Dragon (Peter, Paul and Mary, 1963)
I’ve Lost My Mummy (Rolf Harris, 1963)
A couple of non-PC tunes from the era (1962 actually):
Speedy Gonzales - Pat Boone
Ahab the Arab - Ray Stevens