Most of the father's speech reads like a classic case of a parent trying to teach his child some manners. We certainly get an idea of what his values are and what he views as "superior" behavior. Of course, we're less sure what the speaker, who represents the next generation, thinks about the father's values. We should consider whether the speaker quotes her father to continue his customs, or to criticize them.
The father is not just, literally, the speaker's father, but he symbolizes the past, and all of its old-fashioned values. "Silence" shows how difficult it can be for the current generation to cast off this past.
The father is like the cat's prey, with his quote hanging out of the speaker's mouth just like the mouse's tail hangs out of the cat's. This poem is the speaker's attempt to gobble him up, so that she can finally become the superior person.