IIRC Parallax and Normal maps are essentially alternatives, so doing one rules out the other. If all is equal, a Normal map should be able to achieve anything a Parallax map can, but the latter is simpler, and so quicker to render, making it a better choice for brick and stone walls with nice sharp mortar lines (except that worn stone looks better rounded with Normal maps).
no and no.
Parallax and normal maps work together, both parallax and normal maps render on the same mesh. they work differently, so one or the other doesn't quite and can't do what the other can. parallax is a sorts of displacement map in construction and function, displaces the UV. so you get this scrolling effect in the textures, making it look as if you can almost see around the details in the map.
sorta like a bump map, like a normal map is, but its really a type of displacement map, if you get me. they both do the same thing, ie trick you into thinking the surface of an object has more detailed geometry than it does, but in its functions its really different. And they are expensive as hell to render, not quicker at all.
Parallax is really good for large flowing displacement, or the blunt out line of brick shapes. not lots of sharp detail.