Visualization of biological mesoscale models provides a glimpse at the inner workings of living cells. One of the most complex components of these models is DNA, which is of fundamental importance for all forms of life. Modeling the 3D structure of genomes has previously only been attempted by sequential approaches. We present the first parallel approach for the instant construction of DNA structures. Traditionally, such structures are generated with algorithms like random walk, which have inherent sequential constraints. These algorithms result in the desired structure, are easy to control, and simple to formulate. Their execution, however, is very time-consuming, as they are not designed to exploit parallelism. We propose an approach to parallelize the process, facilitating an implementation on the GPU.