Does a father's statement to his son: "I will hit you if you play too much and are late for prayer," constitute an oath (yameen) that necessitates expiation if the father does not fulfill it, even though he did not mention the name of Allah, but there is emphasis through the threat?

The oath for which expiation is obligatory is the affirmation of something not obligatory by mentioning the name of Allah or one of His attributes. What was mentioned is not an oath, but rather an emphatic warning. Breaking it does not entail sin, expiation, or blame; on the contrary, its doer may even be praised.