By the way, I think the part about the poem being published in 1863 is incorrect. I saw it in one of the many later editions of Water Babies, which I found at my local library, but it couldn't have been in the original 1863 edition. It was probably first published posthumously, in 1877, in a memoir by his wife. You'll see there (just above the text) that she says he wrote it during a severe illness he had while in Colorado during July of 1874. He was visiting the US on a lecture tour. And she says that it was the last thing he ever wrote. He apparently never fully recovered, and died the next year.

And its being written in Colorado raises the question of whether it might have been based on a true story he heard while in the US. The story seems too bizarre to have been made up, which is the way I always felt about the ballad of Omie Wise before I learned that it was in fact based on a true story. There are a lot of places in the US called Coulter, including a creek, lake, and mesa in Colorado and a brook in New York, though there are apparently none in England (the parody you cited comments on that). And a horse race might well have been held in a meadow (lea) at Coulter Brook or at one of the towns or lakes of that name.