Many textures require complex models to describe their intricate structures. Their modeling can be simplified if they are considered composites of simpler subtextures. After an initial, unsupervised segmentation of the composite texture into the subtextures, it can be described at two levels. One is a label map texture, which captures the layout of the different subtextures. The other consists of the different subtextures. This scheme has to be refined to also include mutual influences between textures, mainly found near their boundaries. The proposed composite texture model also includes these. The paper describes an improved implementation of this idea. Whereas in a previous implementation subtextures and their interactions were synthesized sequentially, this paper proposes a parallel implementation, which yields results of higher quality.