The Tora instructed us to stick to the truth and stay away from lies. We have already seen, however, that in some exceptional cases the Rabbis acknowledged that it was permitted or even recommended to withhold the true facts (see here).
The Rabbis also considered that when a person is faced with an embarrassing situation, he is also allowed to change the story. For example, if a person goes to an hospital to see a doctor or a therapist and he does not want people to know about his or her specific problem, and he meets his neighbor in the hospital, he is allowed to withhold the truth and conceal the real reason for his visit to the hospital.
It is important to remember that all the exceptions we have mentioned so far, i.e., all the cases in which we can depart from the truth, refer to situations where our narrative does not affect, damages or hurts somebody else. If there are two partners, in a business, for example, and one of them made a mistake that somehow affects the company, he can not conceal his mistake from his partner adducing embarrassment.
Similarly, a son or a daughter should not lie to his or her parents (or teachers) in an attempt to avoid embarrassment. Adults are supposed to know (and accept!) that a son or a student might fail or make a mistake. And adults are there to guide and teach him or her to learn from mistakes.
By the way, we see how important is for parents to praise our children when, despite the embarrassment, they dare to tell us the truth. When children find a mature, confident and empathetic adult that helps the child processing his embarrassment in a positive way, learning for the future, in all likelihood, that child will have no reason to hide his mistakes from his parents, and he will be in a better position to avoid repeating them in the future.