Does a brother share in the sin if he gives his old phone to his sister, while being certain that she will use it to watch TV series that contain idol or cow worship, listen to music, and commit other sins?

You should continue advising your sister to abandon backbiting, listening to music, and watching TV series that contain corrupt beliefs, music, and illicit relationships, as these are objectionable acts and a waste of time, about which a person will be questioned on the Day of Judgment, as stated in the Hadith: "The feet of a servant will not move on the Day of Resurrection until he is asked about his life and how he spent it, about his knowledge and what he did with it, about his wealth and how he acquired it and how he spent it, and about his body and how he wore it out."

Regarding the ruling on watching television, Sheikh Ibn Uthaymeen mentioned that it is divided into three categories: 1. Watching news, religious discussions, and cosmic observations: There is no harm in this. 2. Watching seductive series and criminal acts: This is forbidden and impermissible. 3. Watching what wastes time and is not explicitly forbidden, but involves a degree of doubt: One should not waste time on it.

If your parents bring you a new phone and stipulate that your old phone be given to your sister, and it is highly probable that she will use it for forbidden things, then there is no blame upon you, because you did not give her the phone; rather, your parents are the ones who will dispose of it. You can inform them that you will return the phone to them.