How authentic is this supplication for a parent, is it from the Sunnah, and is it permissible to supplicate with it?

The meaning of the aforementioned supplication is sound, and there is no harm in supplicating with it. However, this specific supplication is not among the transmitted Sunnah. It is permissible for a Muslim to supplicate for whatever good they desire in this world and the hereafter. It is preferable to supplicate with the supplications mentioned in the Quran and Sunnah, such as: "Our Lord, forgive me and my parents" and "My Lord, have mercy upon them as they brought me up [when I was] small."