Often, in living cells different molecular species compete for binding to the same molecular target. Typical examples are the competition of genes for the transcription machinery or the competition of mRNAs for the translation machinery. Here we show that such systems have specific regulatory features and how they can be analysed. We derive a theory for molecular competition in parallel reaction networks. Analytical expressions for the response of network fluxes to changes in the total competitor and common target pools indicate the precise conditions for ultrasensitivity and intuitive rules for competitor strength. The calculations are based on measurable concentrations of the competitor-target complexes. We show that kinetic parameters, which are usually tedious to determine, are not required in the calculations. Given their simplicity, the obtained equations are easily applied to networks of any dimension. The new theory is illustrated for competing sigma factors in bacterial transcription and for a genome-wide network of yeast mRNAs competing for ribosomes. We conclude that molecular competition can drastically influence the network fluxes and lead to negative response coefficients and ultrasensitivity. Competitors that bind a large fraction of the target, like bacterial sigma(70), tend to influence competing pathways strongly. The less a competitor is saturated by the target, the more sensitive it is to changes in the concentration of the target, as well as to other competitors. As a consequence, most of the mRNAs in yeast turn out to respond ultrasensitively to changes in ribosome concentration. Finally, applying the theory to a genome-wide dataset we observe that high and low response mRNAs exhibit distinct Gene Ontology profiles.