To save face is to try to regain favorable standing after something embarrassing has happened, so in this context speaker is referring to the people who would tell a child that his mother was dead to avoid embarrassment.
Consider the statement "Are those people-the kind of people who would tell a child that his mother was dead just to avoid embarrassment-are they the ones you want to help?". Is the statement "Are those people-the kind of people who would tell a child that his mother was dead just to save face-are they the ones you want to help?" false? Yes/no/unknown