Respectfully disagree. You're essentially arguing that they're not politically motivated in staying at/leaving their job. That's fair. Nevertheless their actions translate either support, inaction, or opposition. The implication which MLK argues is that inaction is harmful to the movement.
Which, as your parent seems to not realize, is also a political stance. This dichotomy is exactly what MLK refers to in the Letter from a Bermingham Jail (i.e. positive vs negative peace).