"Father put the cow away" probably started out as a parody of this poem:

From Fireside Thoughts, Ballads, etc., etc. by Claribel [Charlotte Alington Barnard] (London: James Nisbet and Co., 1865), page 67:

SUSAN'S STORY.

Oh mother take the wheel away and put it out of sight,
For I am heavy-hearted, and I cannot spin to-night;
Come nearer, nearer yet, I have a story for your ear,
So come and sit beside me, come and listen, mother dear;
You heard the village bells to-day, his wedding bells they were,
And Mabel is his happy wife, and I am lonely here;
A year ago to-night, I mind, he woo'd me for his bride,
And who so glad at heart as I that happy easter-tide.

But Mabel came among us, and her face was fair to see,
What wonder was it, mother, that he thought no more of me;
When first he said fair words to her, I know she would not hear,
But in the end she listen'd, could she help it, mother dear?
And afterwards we met, and we were friendly all the same,
For ne'er a word I said to them of anger or of blame;
Till both believed I did not care, and may be they were right,
But mother put the wheel away, I cannot spin to-night.