Is it permissible for the questioner to be paid for managing his father’s properties and collecting their rents, especially since his siblings do not help him and his father does not appreciate his efforts financially?

You are not permitted to take anything from your father's rental income without his knowledge. Otherwise, seek your reward from Allah for serving him, for that is among the best deeds. Do not let the conduct of your siblings deter you from this. Nevertheless, if this work impedes your own interests and causes you to be unable to fulfill your own obligations, then you are not required to do it.