  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 magnetic
field within a Cartesian framework that captures the essential features of MHD
oscillations in flux tubes. We derive a Mathieu equation for the Lagrangian
displacement of the interface and analyse its properties, identifying two
different instabilities: a Kelvin-Helmholtz instability and a parametric
instability involving resonance between the oscillatory shear flow and two
surface Alfv\'{e}n 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 extract energy.
When applied to flows with the characteristic properties of kink waves in the
solar corona, both instabilities can grow, with the parametric instability
capable of generating smaller scale disturbances along the magnetic field than
possible via the Kelvin-Helmholtz instability. The characteristic time-scale
for these instabilities is $\sim 100$ 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.
