According to the Dr Demento Official Website, the song was recorded on Allan in Wonderland, Warner Brothers W/WS 1539 (1964) and The Best Of Allan Sherman, Valiant VS-112 (1965) (U.K.)
Lyrics from the above website

Skin
(Parody of "Heart")
By Allan Sherman

You gotta have skin.
All you really need is skin.
Skin's the thing that if you got it outside,
It helps keep your insides in.

It covers your nose,
And it's wrapped around your toes.
And inside it you put lemon meringue,
And outside ya hang your clothes.

Skin is what ya feel at home in.
And without it, furthermore,
Both your liver and abdomen
Would keep falling on the floor.

(And you'd be dressed in your intestine)

A Siamese twin
Needs an extra set of skin.
When the doctor knows that you're feeling sick
Where does he stick his needle in?
In the end of your skin.

All your friends and all your kinfolks,
Whether poor or whether rich,
They have all got lots of skin, folks,
It's convenient when they itch.

(Nothing can match it when ya scratch it)

It fits perfectly.
Yours fits you, and mine fits me.
When you're sitting down, it folds and looks grand.
And then when you stand, it's where it's been.
Ain't ya glad you've got skin!

When you were just a little baby, why your skin fit fine.
And it still is gonna fit you when you're six foot nine.
So whether you're fat, tall, big, small, chubby or thin,
Ain't ya glad you've got skin!