Normals mapping is only a type of *texture*... Normals mapping can be used for bumpmaps (what you're talking about) -- parallax mapping -- distortion -- whatever.
What's confusing a lot of people is that since Normals Maps are so commonly used for bumpmapping -- they kind-of taken on a synonymy.
As for "Parallax Mapping" -- it's IN UnrealTournament3.
http://phalanx.planetunreal.gamespy..../examples2.jpg
The "Bump Offset" node in the materials editor -- determines whether or not you want a standard bump or a parallax bump. Parallax bumps can take on a heightmap (greyscale) to determine how "high" a texel is on the polygon... I think they can also take a normalmap (colour) but don't quote me on that...
Let me explain.
Normal mapping is used to transform the angle of the normal on the surface of a polygon. Traditional lighting methods used in previous generation games only looked up the polygon normal and the vertex normal. However this meant that the shadowing quality was dependent on the number of vertices and polygons. Hence why we say massive increases in polygon counts over the years, where as now we are see the reduction. We now use per-pixel lighting, so instead of stretching a gradient over a polygon, we evaluate each light value per pixel now. We evaluate the incoming light vector together with the surface normal vector, altered by the normal map. The illusion of depth is provided by the lighting created on an other wise flat surface.
Parallax mapping attempts to create the illusion of actual depth of a surface by distorting the texture coordinates of the pixel in relation to the camera vector. This can lead to pretty good results depending on the camera vector (fails at extreme viewing angles). Since you are distorting the texture coordinates, you cannot create new data and thus you merely stretch the texture. This is a very fast method of achieving unlit and lit depth.
Steep parallax mapping is a ray tracing algorithm. It takes the camera vector, and casts a ray against a height map, and pulling out the correct pixel value from the diffuse map. Because we are using a ray cast to figure out what pixel to actually render, it is very accurate. However, the more 'steps' we introduce for the height map the better the resultant image is, but it becomes slow. Reducing the number of steps creates a more pixelated/ mosaic appearance but it is much faster. Because we are doing a ray cast, we can optionally do another ray cast using the direction of the light vector, thus we can have shadows as well. Thus essentially, we have created a portal using the polygon surface, a diffuse map and a height map. This is a very nice effect, but it is very slow. I have some cases where it is a good idea to mix this method with normal mapping (use normal mapping when the distances become larger)