The Kelvin–Helmholtz instability has been proposed as a mechanism to extract energy from magnetohydrodynamic (MHD) kink waves in flux tubes, and to drive dissipation of this wave energy through turbulence. It is therefore a potentially important process in heating the solar corona. However, it is unclear how the instability is influenced by the oscillatory shear flow associated with an MHD wave. We investigate the linear stability of a discontinuous oscillatory shear flow in the presence of a horizontal mag- netic field within a Cartesian framework that captures the essential features of MHD oscillations in flux tubes. We derive a Mathieu equation for the Lagrangian displace- ment of the interface and analyse its properties, identifying two different instabilities: a Kelvin–Helmholtz instability and a parametric instability involving resonance be- tween the oscillatory shear flow and two surface Alfv ́en waves. The latter occurs when the system is Kelvin–Helmholtz stable, thus favouring modes that vary along the flux tube, and as a consequence provides an important and additional mechanism to ex- tract energy. When applied to flows with the characteristic properties of kink waves in the solar corona, both instabilities can grow, with the parametric instability capa- ble of generating smaller scale disturbances along the magnetic field than possible via the Kelvin–Helmholtz instability. The characteristic time-scale for these instabilities is ∼ 100 s, for wavelengths of 200 km. The parametric instability is more likely to occur for smaller density contrasts and larger velocity shears, making its development more likely on coronal loops than on prominence threads.