I always thought it was sung to his mule.

The verse should've been:

My old man's good old man
Washed his face in a frying pan,
Combed his hair with a wagon wheel,
And died with a toothache in his heel.

another few verse I thought of:

I gee to the mule but the mule wouldn't gee, (3 times)
So I hit him side the head with the single tree.

I haw to the mule wouldn't haw, (3 times)
So I broke his back with my mother-in-law.

Art Thieme

^^