The Qur'an gives believers specific words for praying for their parents. They can be said for parents who are alive, and for those who have passed away. Their Arabic is on this page; this narration gives their meaning.

The best known comes in Surah Al-Isra, chapter seventeen, verse twenty-four, which commands gentleness and humility towards parents, and then gives these words: “My Lord, have mercy upon them as they brought me up when I was small.” Many include it in their daily supplication.

The prophet Ibrahim prayed, in Surah Ibrahim: “Our Lord, forgive me and my parents and the believers the Day the account is established.” It is suitable at the end of any supplication. The prophet Nuh prayed more widely, in Surah Nuh: “My Lord, forgive me and my parents and whoever enters my house a believer and the believing men and believing women.” And Surah Al-Ahqaf preserves the prayer of a person reaching maturity, giving thanks for Allah's favour on themselves and their parents, and asking that righteousness pass on to their own children.

Why is praying for parents who have died so emphasised? The Prophet, peace be upon him, said, in meaning: “When a person dies, their deeds come to an end except for three: an ongoing charity, knowledge that is benefited from, or a righteous child who supplicates for them.” That is in Sahih Muslim. A child's sincere supplication is one of the few things that still reaches them. Pair it with ongoing good on their behalf, such as charity, teaching, and keeping their good ties.

Praying for parents does not replace treating them well in life. And there is no set time: after the five prayers, in prostration, and in the last third of the night are all encouraged times, but any time is valid.

One serious question is a parent who died outside Islam. Scholars answer it from the texts, including a report in Sahih Muslim and a verse in Surah At-Tawbah that restrict seeking forgiveness for those who died rejecting faith. For living parents who are not Muslim, praying for their guidance and treating them well is encouraged. For your own situation, ask a qualified scholar. And for the Arabic of these supplications, learn from a teacher or a verified recording.