Barmitzvah observances and celebrations are pretty new in Judaism. They do not appear in the Talmud and begin to take shape in the Middle Ages.
One of the first mentioned is the blessing said by a father on his son’s barmitzvah, Baruch Shep’tarani...meaning, “Blessed be the One who has exempted me from this one’s (ie his child’s) punishment.”
The first halachic source where this appears is from the 14th century, though it refers to a tradition that Rav Yehudai Gaon, who lived in the eighth century, said the blessing the first time his son read from the Torah.
The idea that you publicly express gratitude for now being off the hook for your kid’s punishments makes some people uncomfortable and I believe that the blessing is no longer said in many non-Orthodox synagogues. One could argue, though, that it’s healthy to acknowledge the limits of your responsibility at this point.
Accepting that your child is his/her own moral agent is, the parenting books tells us, one of the keys to raising teenagers.