Is it permissible for a teacher to ask students' parents to search for students for him personally, and not for the academy he works for, or to request links to Muslim groups on social media programs?

The questioner is not forbidden from doing that, unless the academy he works for stipulated that he should not do so. This is based on the principle of permissibility and the obligation to fulfill conditions and contracts, according to the Almighty’s saying: "O you who have believed, fulfill [all] contracts," and the Prophet’s (peace be upon him) saying: "Muslims are bound by their stipulations." This also applies if custom dictates it and is consistently practiced, because what is known by custom is like what is stipulated by condition.