Is it considered a sin to allow the children of some individuals working under my management to perform work on behalf of their fathers, knowing that this is systematically forbidden and unknown to officials, and that the reason for my permission is in appreciation of their old age?

If the intention is to avoid hardship and show kindness to Muslims, especially the elderly, then we ask Allah to include you in the supplication of the Prophet (peace be upon him): "O Allah, whoever takes charge of any matter of my Ummah and makes it difficult for them, then make it difficult for him. And whoever takes charge of any matter of my Ummah and is kind to them, then be kind to him." However, kindness must not come at the expense of work, or in violation of regulations or contracts, as Allah Almighty says: "O you who have believed, fulfill [all] contracts." (Quran 5:1). Nor should it involve transgression against officials or exposing oneself to violations. Therefore, what you did is not permissible.