Is it permissible for a Muslim to move in with one of his parents in their rented apartment with the aim of preserving it after their death and having the lease transferred to him at the same meager rent, knowing that this action is legally permissible? And is this considered an usurpation of the owner's right?

The old lease contract is null and void according to Islamic law, because a lease must be for a known duration. For those who deem a month-to-month contract valid, either party may terminate it at will. This law does not permit what is forbidden nor forbid what is permissible. Contracts that do not specify a duration or are not month-to-month must be terminated, and the leased property returned to its owner with the payment of a fair market rent, unless the owner waives it. It is not permissible to reside in the aforementioned apartment, and your father must return it to the owner. Strive to honor your parents in a manner that does not violate Islamic law. For leases of fixed duration, the majority of scholars hold that they do not terminate upon the death of the tenant but are inheritable, while the Hanafis are of the opinion that they terminate upon the death of the tenant.