I just use Incognito mode on Chrome all the time, even though there is no suspicious activity going on! (Well, aside from being on a furry forum, but that’s not super suspicious, and I’m pretty sure that my parents would be fine with it!) If my parents really wanted to, they could check the incoming traffic for the router, and put it into an analysis program, but since I don’t know how to do it, I doubt they do either! Would they even want to? I wouldn’t!
(Full disclosure: I did at some point wander into a couple of 18+ areas, but I quickly realized that it was a bad idea! I am not going on those things until I’m 18!)