That would be an absolutely awful user experience, unless there was also a way to default to knowing what the user's actual local timezone is without them having to manually pick it from a list of the 38 or so currently in use. I mean, you could try to persuade browser builders and site developers that this new get parameter is a standard that is automatically added to all requests by all browsers, and honoured if the site developers feel like it, but that's kind of messy and effectively doing the same job that request headers were designed to do.
You could have the user create an account which remembers which timezone they picked on signup.