Is the widely circulated story of Abu Yazid al-Bistami authentic, and is it permissible to say, "Arise, my son, for you are more beloved to God than your father," which is mentioned therein?

There is no person by the name of "Abu Ziyad al-Bistami"; rather, it is "Abu Yazid al-Bistami." The phrase "You are more worthy of Allah" (أنت أولى بالله) is permissible and means closer to Allah, similar to the noble Hadith, "Indeed, the people most worthy of Allah are those who initiate the greeting of peace to them." The story mentioned in the message was not found; however, there is a similar story that happened to Abu Yazid al-Bistami when he was young, where he urged his father to perform night prayers after hearing Quranic verses about the Prophet, peace and blessings be upon him, and the Companions, may Allah be pleased with them, performing night prayers. He expressed his desire to pray with him despite his young age.