I disagree. Parents should have the responsibility to know what their kids are doing. Although it's true that gambling is accessible now to most people even kids with online gambling, parents should be able to inform or educate their kids about how to handle gambling and it's effects.
Indeed, parents should monitor what their kids are doing online. Not only gambling is proliferating online but also pornography,violent games,videos etc. Its better to block these sites and have some firewall and check their browser history.Also,if they have a mobile phone,you can check that also, and it is a good policy to them that you must know their password.
I think it is possible to restrict kids in accessing sites especially gambling and pornography. There are apps that can be installed and will block sites that have in connection in both gambling and porn sites. But if the kids started to browse in an internet cafe, there is a possibility that they might end up browsing the sites you are blocking at home. Either from cookies or thru searches and ads that is shown in a social sites.