Force-driven parallel shear flow in a spatially periodic domain is shown to be linearly unstable with respect to both the Reynolds number and the domain aspect ratio. This finding is confirmed by computer simulations, and a simple expression is derived to determine stable flow conditions. Periodic extensions of Couette and Poiseuille flows are unstable at Reynolds numbers two orders of magnitude smaller than their aperiodic equivalents because the periodic boundaries impose fundamentally different constraints. This instability has important implications for designing computational models of nonlinear dynamic processes with periodicity.