Is it permissible for me to kill the birds my father keeps in the house due to my mother, who suffers from allergies, being harmed by them, and my inability to convince my father to remove them or release them?

If a trustworthy doctor confirms that the sparrows are harmful to your mother, strive to convince your father to get rid of them, and seek help from someone who can influence him. If that doesn't work, then it is permissible for you to kill them if there is no other way to get rid of them, because human health takes precedence over animals.