The parents were pleased that Jane was appointed as a teacher at the school.
Literal
Parents [topic-は] Jane [subject-が] that school [possessive-の] teacher [as-に] was-appointed pleased.
任じられる is the passive of 任じる (to appoint) — a formal, somewhat literary verb. て connects the appointment clause to the parents' emotional response (喜んだ). The passive marks Jane as the recipient of the appointment.