Is it permissible for a father to deem singing lawful based on his personal opinion, claiming that musical instruments did not exist during the time of the Prophet, peace and blessings be upon him?

No one is permitted to declare something lawful or unlawful without a legitimate Shar'i (Islamic legal) proof.

Texts from the Sunnah have indicated the prohibition of musical instruments. Among them is the saying of the Prophet, peace be upon him: "There will be people from my Ummah (nation) who will consider lawful illicit sexual intercourse, silk, alcoholic drinks, and musical instruments" (Narrated by Al-Bukhari). And his saying: "Two voices are cursed in this world and the Hereafter: a wind instrument at a time of blessing, and a wailing at a time of misfortune" (Narrated by Al-Bazzar and Al-Diya Al-Maqdisi). And his saying: "Indeed, I did not forbid weeping, but I forbade two foolish, immoral voices: a voice at a time of pleasure, amusement, play, and the flutes of Satan..." (Narrated by Al-Hakim, Al-Bayhaqi, and Ibn Abi Al-Dunya). And his saying: "In this Ummah, there will be land degradation, transformation (of people into animals), and pelting with stones, when singing girls and musical instruments appear, and alcoholic drinks are consumed" (Narrated by Al-Tirmidhi).

Scholars have reached a consensus on the prohibition of using instruments of amusement and musical instruments, with the exception of the tambourine (daf). Among those who conveyed this consensus are Abu Al-Tayyib Al-Tabari, Al-Qurtubi, Ibn Rajab, Ibn Al-Salah, and Ibn Hajar Al-Haytami.

The questioner is advised to be gentle with his father when offering advice.