Turn off Reduce Motion and get over it. Reduce Motion does exactly what the name implies, reduces motion effects throughout the OS. The animations are sped up significantly in 7.1, so that is not an issue, and the parallax effect can be disabled separately when changing wallpaper, so you don't have a battery issue but can still have the animations, which is what the weather effect is.
There you go, problem solved.
While I thank you for the suggestion of enabling motion then manually turning off parallax on a per-background basis, no, that still doesn't address the absurd connection. The weather app is an app, not the OS. Reduce Motion doesn't turn off animation in games or GIFs on web pages or anything else cause that'd be silly. It should just affect app transitions, opening folders, desktop backgrounds.. you know, things which are part of the OS i.e. app-independent.
Also, while I can't say myself if app transitions are faster in 7.1 than they were previously, they're definitely still slower than motion-reduced transitions, at least on a 5C, especially the desktop fly-in after unlocking the phone.
Finally, this is an
accessibility option. I don't get motion-sickness like some people do from those new OS transitions, but that is the main point of this feature. Was weather making people motion-sick? I really doubt it; it doesn't rapidly move your perspective to make you feel falling or flying or anything. It's just a slow, pretty animation.
Is it a big deal either way? Of course not. But it baffles me nonetheless. It's a completely pointless thing to have done.
Today I also learned that if you toggle Reduce Motion back on, your backgrounds revert to parallaxing regardless how you had them set. You must go assign them again after messing with that accessibility option if you want them to remain stationary.