Are parents held accountable for the sins of their children, such as lying and indecencies, while they are under the age of puberty, or is the sin only for negligence, based on the saying of the Chosen One: "It is enough sin for a man to neglect those he is responsible for"?

There is no sin upon the parents for their son's mistakes simply because the mistake occurred, for the Pen is lifted from the child. However, sin falls upon the parent if they neglect to raise their child according to the Sharia (definition) and noble morals, for Allah has commanded His servants to care for their children, protect them from the Fire, teach them, and discipline them. This is in accordance with His, the Almighty's, saying: "O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones," and the Prophet, peace and blessings be upon him, said: "And a man is a guardian over his household and is responsible for his subjects."