The right to murder family members ended in the first century BC, although, even then, they kept a few exceptions. Now, the law said, fathers could only murder their sons if they’ve been convicted of a crime. (listverse.com)
Which law? Is there a source to backup this (at this date or later on)?
I read several articles, including:
This interesting study (which concluded that the practice was abandoned after a while: "While the paterfamilias could kill, he could not get away with it"). But the study doesn't mention the law(s) this quote is talking about.
Stephen Tempest answer on Quora also think this "right" finally ended (but he doesn't provide a source)
However, it seems that at least by the beginning of the Principate [around 30 BCE] the theoretical power of life and death held by a father over his children was very rarely used, and it was regarded as horrifying when it was.
Seneca describes a case during the reign of Augustus where a man named Tricho had his son flogged to death for some crime or other; but an angry mob then lynched Tricho himself, right in the middle of Rome.
In the second century CE, Emperor Hadrian discovered that a father had killed his own son after discovering that the son had been having sex with his stepmother (the father's new wife). However, Hadrian stated that "A father's power should be based on affection, not cruelty", and ordered that the father himself be punished by banishment. It then became the law that a father could not kill his child unless the child had first been convicted by a regular court of law.
Selling children into slavery also seems to have fallen into disuse as a custom by the time of the Principate, except in the purely token form of the ceremony of emancipation, and also sometimes in the case of children charged with a crime, where the child was handed over to the victim in recompense. It may have still happened occasionally in cases of desperate poverty, and with newborn children.
After the Empire converted to Christianity the concept of patria potestas lingered on in some legislation, but by then it was very much changed from its earlier form.