No. That is not grounds for Temporary Protected Status. Only citizens of certain countries which had natural disasters of such significant extent as to preclude an immediate return to their country of citizenship may be granted such temporary permission to remain. The citizens of such a country had to be in the US on the date of the disaster, and apply for such status within a limited time frame thereafter. Furthermore, this is always temporary, and the persons involved are not eligible for any sort of long-term or permanent status such as legal permanent resident. They need to return to their country of citizenship prior to expiration of the TPS declared for their particular country.
Anyone brought in illegally as a minor child by his/her parents already has a means to change that situation without penalty. If they return to their country of citizenship before their 18th birthday, they will not be penalized for their parents' illegal acts, and may later apply for any type of visa for which they qualify. However, as of their 18th birthday, it becomes their choice - as an ADULT! - to violate the law, and they immediately become subject to arrest, detention, deportation, and bar to reentry. There is NO way to "legalize" an illegal alien, not school, marriage, procreation or anything else. If they do not take their one and only chance to make things right, they have NO alternative but to return to their country of citizenship and be barred from reentry, usually for 10 years due to the duration of their illegal presence.
Children have a way out of the mess their parents put them in. Adults go home - no exceptions, no excuses. And they can no longer blame their parents for their own lawbreaking.