Weyl semimetals typically appear in systems in which either time-reversal (T ) or inversion (P ) symmetry is broken. Here we show that in the presence of gauge potentials these topological states of matter can also arise in fermionic lattices preserving both T and P . We analyze in detail the case of a cubic lattice model with π fluxes, discussing the role of gauge symmetries in the formation of Weyl points and the difference between the physical and the canonical T and P symmetries. We examine the robustness of this P T -invariant Weyl semimetal phase against perturbations that remove the chiral sublattice symmetries, and we discuss further generalizations. Finally, motivated by advances in ultracold-atom experiments and by the possibility of using synthetic magnetic fields, we study the effect of random perturbations of the magnetic fluxes, which can be compared to a local disorder in realistic scenarios.