YES, this is an example of an appeal to ETHOS.
We speak of "ethos," "pathos," and "logos," based on Greek terms that were already discussed by early philosophers such as Socrates, Plato, and Aristotle. Aristotle, in particular, used such terms to refer to ways in which we appeal to others to convince them of something.
Logos ( λόγος) is the Greek word for "reason" or "logic" (as we bring their word into English. An appeal to logos is an appeal to what is rational and thoughtful.
Pathos (πάθος) is the Greek word for "suffering" which in turn becomes a word to describe emotions -- because emotions can make us ache inside as we experience the ups and downs of life. An appeal to pathos is an emotional appeal.
Ethos (ἦθος) is the Greek word for "character" which has to do with one's morals -- right attitudes and right behaviors. An appeal to ethos is an appeal to do the right thing, based on trust in authority figures (such as a father, in the example listed) or based on what makes for a flourishing and well-lived life. Aristotle's work on Ethics focused on that idea of the kind of life that is best for a human being to live.
So in the example, while there is some appeal to emotions (the father's love for the son), the main appeal is to the authority of the father and the urging to do the right thing, which is an ethical appeal.
