Modeling and shaping how information spreads through a network is a major research topic in network analysis. While initially the focus has been mostly on efficiency, recently fairness criteria have been taken into account in this setting. Most work has focused on the maximin criteria however, and thus still different groups can receive very different shares of information. In this work we propose to consider fairness as a notion to be guaranteed by an algorithm rather than as a criterion to be maximized. To this end, we propose three optimization problems that aim at maximizing the overall spread while enforcing strict levels of demographic parity fairness via constraints (either ex-post or ex-ante). The level of fairness hence becomes a user choice rather than a property to be observed upon output. We study this setting from various perspectives. First, we prove that the cost of introducing demographic parity can be high in terms of both overall spread and computational complexity, i.e., the price of fairness may be unbounded for all three problems and optimal solutions are hard to compute, in some case even approximately or when fairness constraints may be violated. For one of our problems, we still design an algorithm with both constant approximation factor and fairness violation. We also give two heuristics that allow the user to choose the tolerated fairness violation. By means of an extensive experimental study, we show that our algorithms perform well in practice, that is, they achieve the best demographic parity fairness values. For certain instances we additionally even obtain an overall spread comparable to the most efficient algorithms that come without any fairness guarantee, indicating that the empirical price of fairness may actually be small when using our algorithms.