Is Safari allowing extensions to be installed by stray clicks in a web page? Sometimes when I go to visit my parents, they ask me to fix their Macbook because something is wrong with Safari. The find that there is an extension installed and that it has set the home page in the browser. Unfortunately I don’t remember the name of the last one but I think it was “Smart Search” or “Safe Search” or something like that. 

I know how to uninstall an extension, but I’m wondering how it gets installed in the first place. I think my parents know enough about computers to not download and install random programs of the internet so I suspect they are getting tricked into clicking something in a web page. But, is that possible? Can a website install an extension on the sly like that? It’s never happened to me.

They're running macOS High Sierra 10.13.5, Safari 11.1.1.
It’s far more likely they installed software that says it’s for “playing videos” or “flash player malware / adware” or just installing Java which loads cruft now a days unless you carefully read and correctly opt out.

If you run Malware Bytes anti Mac - what specific malware and adware are your parents getting in to? That would give you specific bad actors and then be able to target specific behaviors or training needed.

You could try installing a locked down browser like Brave, but I’m guessing they’ll want something that has un-desired add on and you’ll either need to outsource the cleaning or remove the admin account. Even iPads get hijacked with bad ads that pop up scam messages. It’s a never ending war of social engineering that you’re fighting here.