You’re right, of course. But I wonder if the parent is aware other people can detect this sort of thing.

If you can't detect it in some cases, how would you know?

Obviously you can't, and the end result is that this dramatically amplifies the effect of catching someone out in an insincerity. The stronger your belief that someone is truthful, the larger a portion of their actions you need to re-examine once catching them out in a significant lie, and the longer it will take for you to re-build your original belief that they were generally truthful.

(Statistics isn't my strong suit but I believe this is perfectly in line with modeling a person's trustworthiness as a Bayesian process?)