The Shi arrangement is a famous hyperplane arrangement, and its regions are in bijection with parking functions; there are two somewhat-different proofs of this bijection, one by I. Pak and R. Stanley and one by C. Athanasiadis and S. Linusson. The first bijection can be realized by a model involving orientations of certain mixed graphs. The first goal of our project is to establish this realization thoroughly (i.e., mathematically sound). The second goal is to generalize it to signed graphs, in the hope that this will shed additional light onto the main result in this recent paper by K. Meszaros; see also this even more recent paper of D. Armstrong, V. Reiner, and B. Rhoades.