It's an effect the creator can choose when creating the wallpaper.
The effect is called "Camera parallax" and you can find it here when creating a wallpaper:

First off: this only works on scene wallpapers!

Generally you can't edit wallpapers that you downloaded from the workshop. However, I found a way to disable the parallax effect on downloaded workshop wallpapers by editing a specific file in your steam directory.

First, browse to <Steam Install Directory>/steamapps/workshop/content/431960
431960 is Wallpaper Engine's Steam app id. In this folder, you can see all the wallpapers you have downloaded. Each one is in a folder and all the folders names are just some numbers (id of the workshop item).

Now you gotta find your wallpaper that you want to edit.
There's two ways to do this:

1. (This is the way I recommend) Go to the workshop page of the wallpaper you want to edit and then right click and select "Copy Page URL" and then paste that to somewhere. It'll look like this:
   You'll want to remember that number at the end. This is the name of the folder you'll have to go into.
   
2. (Not recommended) Just look through all the folders if you're only subscribed to a few wallpapers. You can find your wallpaper by matching the preview.jpg file in the folder with the thumbnail of the wallpaper you're searching for. Also look out for a scene.pkg file because that way you know this is a scene wallpaper.

Now that you have found your wallpaper folder, open the scene.pkg file in this folder in a text editor.
You can use windows notepad for this but I'd recommend to use Notepad++[notepad-plus-plus.org]

After opening the file, press Ctrl + F to search, type in cameraparallaxamount and hit search.
You should find a bit of text that looks like this:



Now you'll want to change the number after cameraparallaxamount to a 0. Don't change anything else or your wallpaper will most likely stop working.
Attention: It is important that the file has the same length after editing. So if you find the number 1 there, you can just change it to a 0. But, if it's something like 1.4200000000001, then you have to change that to 0.0000000000000 (same length!)
After you changed it, the line should look something like this:


That's it. Now just save the file, go into Wallpaper Engine and select another wallpaper and then switch back to the one you want so it reloads the scene.pkg file. Now your wallpaper will display like normal but the parallax effect is gone.

as PinkGirl78 pointed out in the comments, you can do the following to stop web wallpapers from moving:

(this is directly copied from their comment)

If you want to do something similar to Web-based wallpapers, you can do something similar to the index.html you'll find in the content directory.

Find a line such as:
document.addEventListener( 'mousemove', onDocumentMouseMove, false );

The key phrases are 'addEventListener' and 'mousemove' (the rest may vary a little).

Remove it from the code by adding a `//` to the front.
//document.addEventListener( 'mousemove', onDocumentMouseMove, false );
Your wallpaper should now no longer follow the mouse. YMMV.