All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The evidence is based on what is in the Sharee'ah and not based on the actions of the people.
Therefore, if the children of some preachers (callers to Islam) or the preachers themselves commit something forbidden or disliked, then this does not render these matters permissible.
Preachers should be role models for the people, and should be the farthest of all people from committing what is forbidden or indulging in some disliked matters, so that Da’wah will not be blamed, and so that the religion will not be disparaged because of their mistakes. This especially so at this time and age when war is raged by the enemies of Islam against Islam, and there are so many people who conspire against it and seize every error to defame it.
There is no doubt that it is not permissible to publish forbidden pictures on Facebook or other places, such as pictures of women who do not wear Hijaab.
In addition, enabling children to access the Internet without supervision is a means that could corrupt them. Even adult children should be advised, guided and prevented from what is forbidden, if they ever commit it.
The parents should fear Allah and be keen on cultivating their children and nurturing them in a righteous manner, and protecting them from anything that could corrupt their religion or conduct.
Allah Says (what means): {O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones, over which are [appointed] angels, harsh and severe; they do not disobey Allah in what He commands them but do what they are commanded..} [Quran 66:6]
For more benefit, please refer to Fatwa 373993.
Allah knows best.