Is it permissible for my father to take the incentives of absent employees as compensation for his extra work, knowing that he performs their work to ensure the continuity of the government department and because he needs the job, or is it better for him to resign?

It is not permissible to cover for absent employees or to perform their work in collusion with them, because their actions are forbidden (haram) and their salaries are forbidden (haram). The official who covers for them must be alerted to thefulness of his action. If he does not desist, then it is necessary to inform the responsible authorities. It is not necessary to leave the job if the officials have been advised and forbidden from this evil (munkar).