"...where the deer and the WHAT play??????????"

"Home, home on the range, where the deer and the antelope play."

No antelope ever lived in the wild in North America. Some have been imported into zoos, here. That's it.

An animal called a pronghorn is native to America, but it is called "pronghorn" to focus-in on the characteristic which distinguishes them from antelope -- prongs, or branches, on the horns.

So, the song got it wrong.