Blog: "I'm a Material Boy"

[Thomas]

Normalmapping changes the shading, that is how how lights affects a surface. A normalmapped surface without any lights on it does not look anything different.
Parallax offsets the texture uv coordinate (as explained in blog) and this means that it changes the look of a surface even if it is unlit. Parallax also offsets the point from where the normalmap information is gotten (normalmap uses same uv as color-texture) and therefore parallax indirectly affects the shading too.