Analytic and numerical results on the linear theory of a new instability driven by shear in the parallel velocity v∥′ will be presented. This instability exists in the presence of parallel velocity shear and field line curvature. It also requires either parallel viscosity—the full Braginskii stress tensor [S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205]—or parallel compression. The mode exists in an electromagnetic version, where it can enhance the growth rate of an unstable resistive magnetohydrodynamic (MHD) mode or cause an otherwise stable resistive mode to grow. In its electromagnetic form it is global (matches to an ideal MHD outer region) and can influence modes responsible for disruptions; it can also lead to anomalous transport by stochastic field lines as well as by E×B advection. The instability also exists in an electrostatic form for short wavelengths. For most reasonable edge plasma parameters the effect of parallel compression dominates that of parallel viscosity. The properties of this mode, specifically its growth rate and localization width, are compared with those of the usual v∥′ mode [N. D’Angelo, Phys. Fluids 8, 1748 (1965)]. The importance of this mode in edge fluctuations and on more global resistive MHD is discussed.