Hello
To your question.
The fog is made like a character, but in 154x158-(468x636). And directly in RPG Maker you just set the speed,route, frequency, priority and trigger as parallel.
But you can also create fog that works like parallax, and you need to install a Foreground plugin.I used this type of parallax in my older sets.
In short:
I created the fog as an image in Black.Ink. Then I put it in a frame in Pixelmash and in that I create an animation that has 3 frames. Later I put it in a compact frame that RPG Maker supports.