Parents should be, as should everyone else who is involved or engaging within the child. This is the most thinly-veiled excuse I've seen as pawning off responsibility or any form of issues that may arise; nobody is saying you're required to babysit.
How it is a 'thinly-veiled' excuse? I fail to see how asking parents to keep their children in check as an excuse. Parents and their offspring are responsible for what they consume on the internet. People have access to more tools than when the internet was brand-spanking new. I do agree that everything CANNOT be policed, but most of it can be with parental controls, changing wifi/network passwords, blocking adult-like websites from being accessed, etc. Parents should be at least a bit aware of what their child is doing on the internet.

If you're RP'n inappropriately with a child, I can more than assure you, the law would vehemently disagree and say you have a large responsibility of refraining from such activities and routinely exercising caution.
I agree to a certain extent. However, how can one know one is a "child" when people can lie about their age? People assume you're over 18+ when entering the venues, so you have to give these roleplayers a bit of breathing room. Fake their entire life because of being anonymous on the internet? Yeah, it might sound like a lame excuse, but you know the stories of adults sending inappropriate pictures to what they assume is another adult. On an adult website, but turns out the recipient is a minor. Which isn't fair to the adult, because they were misled. But I'll stop there.

gain, the law would disagree on who has the responsibility, especially if something does occur. Adults should behave like adults, and that includes exercising caution when being involved with minors. Doing inappropriate things will be as much of the persons' fault as it is the parents' lack of caution. You are all equally responsible.
Again, a lot of adults don't behave like "adults". This line is very subjective of who you talk about and to. But yes, I do agree with, if you find the other is underage, no inappropriate situations should be occurring between the two parties. If it comes to the point where actual inappropriate pictures are being sent/swapped, messages? Yes, get the law and police involved. Each country has its' own different laws concerning these situations, so it will differ.

We can agree to disagree and that's fine if you don't like my viewpoints and opinions. I don't claim to be an expert or have correct factual information, so we can leave it at that, no?