Hi Matthias!
Thanks for reading the pocket guide!
I think browsers are well equipped to handle heavyweight animations and even parallax. Browsers have gotten much more performant with pretty much every kind of animation in the last few years. It's still possible to push their limits of course, especially with webGL and such, of course.
In most cases I've seen, parallax preforms poorly more because it was implemented poorly. It's possible to make performant parallax animation but only if that's a priority, and it doesn't always seem to be, sadly. Paul Lewis has written a lot about how to make parallax performant, developers.google.com/web/updates/2016/12/perform… and his other articles on the topic are worth a read.
That's not to say that parallax should be everywhere on the web. It's a pretty heavy-handed effect and if it doesn't fit the context or the content even the most performant parallax isn't going to be a good experience for your audience.