Texture mapping is an increasingly popular technique in molecular modeling. It is particularly effective in representing high-resolution surface detail using a low-resolution polygonal model. We describe how texture mapping can be used with parametric molecular surfaces represented as expansions of spherical harmonic functions. We define analytically the texture image and its transformation to a parametric surface. Unlike most methods of texture mapping, this transformation defines a one-to-one correspondence between the surface and the texture; texture coordinates are derived from the location of the surface point and not from physical properties at the surface point. This has advantages for the interactive visualization of surface data. We control the interactive response time by lowering the resolution of the polygon mesh while retaining the high-resolution detail of the texture, or we can lower the resolution of the texture image with the same polygonal model. By using a well-defined convention for texture coordinates, we can use the same image for the original surface or its parametric representation, and we can rapidly switch between images that represent different surface properties without recomputing the texture coordinates. Parametric surfaces allow new flexibility for the visualization of molecular surface data.