My Sister, My father, and I were praying and it was a silent prayer. I was silently reading Surat- Al-Fatiha and My Dad went in Ruku but I was not done reading Fatiha yet. Because he was leading the prayer I went in to Ruku also. Is this wrong and does it invalidate prayer?
Depending on the maddhab you are following, each maddhab has a different rulling about reciting Surat- Al-Fatiha.
Below is to conclude for each maddhab :
Hanafi Maddhab : It is Makruh Tahrim (abominable) on the makmum to recite Surah al-Fatihah behind the imam, either in the sirriyyah (silent) or jahriyyah (loud) prayers, but the makmum prayers remain valid.
Malik Maddhab : The recitation of Surah al-Fatihah is not absolutely obligatory for the makmum either in the sirriyyah (silent) or jahriyyah (loud) prayers, but it is sunnah for the makmum to recite it in the sirriyyah (silent) prayers. Ibn al-Arabi from the Malikiyyah scholars made it obligatory for the makmum to recite it in the sirriyyah (silent) prayer.
Syafie Maddhab : The recitation of Surah al-Fatihah is obligatory as a pillar of the makmum absolutely. The sunnah is recited during the imam's saktah unless it is feared that the imam will not recite another surah, recite a short surah, or continue to bow, then it is obligatory without makruh to recite al-Fatihah together with the imam.
Hanbali Maddhab : The recitation of Surah al-Fatihah is not obligatory for the makmum absolutely but the sunnah for the makmum is to recite it in the sirriyyah (silent) prayer and during the saktah of the imam. There is a narration from Imam Ahmad that the recitation of al-Fatihah is obligatory (on the makmum) in the sirriyyah (silent) prayer.