I read the question different and thought the OP had already determined that the sale did not include any clause for joint tenancy or for the father to retain any interest, which may cause the gift to be incomplete.  I'm not sure without doing further research whether the mere fact that the son permitted the father to remain in that home, without any right that is legally enforceable, would render that an implied life estate arrangement.

@TaxGuyBill But that is a good question you have raised and is worth considering.