Summary: A group of stations in parallel is considered, where the input processes to the various stations are stochastically dependent nondecreasing Lévy processes and the release rates are deterministic linear flows. It is shown that the model of a tandem fluid network, with dependent or independent Lévy inputs to the stations, can be seen as a special case of this construction. In addition to other natural applications, the joint stochastic structure of the workload processes of the different classes in a preemptive resume priority \(M/G/1\) queue can be viewed as a special case of the model considered. Using martingale and regenerative arguments, certain steady state characteristics are studied.