I 100% agree that parents are responsible for their children. But, we have got to recognize that technology advancement has outpaced a good portion of our parents. And, let's not forget about the numbers of children that are being raised by their grandparents.
We are having this discussion on a tech site and thus it's a safe assumption that many involved here are most likely more tech savvy and educated than the general populace. My children are 7 and 11 years old and I can assure you that many of our friends who are also parents have no clue what is out there and the ways in which today's modern devices grant access or provide an avenue for behavior that they do not want there children to engage in.
Fortunately, some children are still immature and not yet wise enough to cover their tracks. For example, a close friend's 11 year old son looked up pornography on his mothers cell phone and didn't know how to erase the browser history or simply cover his tracks and thus... busted! What was his mother's reaction, "I had no idea that he could access stuff like that so easily!?!"
To say, "parents should be responsible" is shortsighted because it's asking people to plan around factors of which they are ignorant. The solution to this parental challenge is more education for parents. Our parents need to have places they can go to learn about the potential dangers of social media and the internet. They need to be trained on how to use parental controls effectively and taught what signs to look for in their children's behavior.
As for the comments about "kids being kids" and this being nothing new, well, yes and no. Human nature is still human nature and youth will be curious about sexuality, but the open access and availability is different in today's world and that is changing the views of younger generations. I'm a professional therapist and the work I do with young people reflects a marked difference in attitudes towards sexuality as opposed to previous generations. As another poster has already mentioned the advent of social media and everyone having a camera has built an expectation in younger generations that sexuality will be captured in media format and shared. There is also less of an emphasis on exclusivity in relationships. One young girl I worked with recently explained how her boyfriend expected her to give sexual favors to his friends just to "help them out" because they didn't have a girlfriend at the time. Pornography exposure has greatly changed the expectations of young men in the bedroom and furthered the stereotypical view that a partner is there to simply be used for your own satisfaction. Teens I've worked with have explained to me that there is a lot of pressure on girls to use apps like Facetime or Snapchat to provide real life porn to peers or significant others so they can "take care of their needs". For example, "I can't find anything online that interest me at the moment Sally, why don't you make me a quick video so I can do my thing..."
Is everything I've mentioned Snapchat's fault and should they be held responsible, no. But, they should do their part to have adequate warnings, age restrictions, and not be serving explicit content that is not searched for. There are numerous programs like Snapchat that can be used irresponsibly so I don't blame them. But, likewise, the solution is not simply to blame parents either. It's a complex issue and since it's a morality issue many will come down on different sides of the topic.