Metal halide perovskites exhibit a materials physics that is distinct from traditional semiconductors. While materials such as CH3NH3PbI3 are nonmagnetic, the presence of heavy elements (Pb and I) in a noncentrosymmetric crystal environment result in a spin splitting of the frontier electronic bands through the Rashba-Dresselhaus effect. We show, from a combination of ab initio molecular dynamics, density-functional theory, and quasiparticle GW theory, that the nature (magnitude and orientation) of the band splitting depends on the local asymmetry around the Pb and I sites in the perovskite structure. The potential fluctuations vary in time as a result of thermal disorder. We show that the same physics emerges both for the organic-inorganic CH3NH3PbI3 and the inorganic CsPbI3 compound. The results are relevant to the photophysics of these compounds and are expected to be general to other lead iodide containing perovskites.