  We investigate a perturbation problem for the three-dimensional compressible
isentropic viscous magnetohydrodynamic system with zero resistivity in the
presence of a modified gravitational force in a vertical strip domain in which
the velocity of the fluid is non-slip on the boundary, and focus on the
stabilizing effect of the (equilibrium) magnetic field through the non-slip
boundary condition. We show that there is a discriminant $\Xi$, depending on
the known physical parameters, for the stability/instability of the
perturbation problem. More precisely, if $\Xi<0$, then the perturbation problem
is unstable, i.e., the Parker instability occurs, while if $\Xi>0$ and the
initial perturbation satisfies some relations, then there exists a global
(perturbation) solution which decays algebraically to zero in time, i.e., the
Parker instability does not happen. The stability results in this paper reveal
the stabilizing effect of the magnetic field through the non-slip boundary
condition and the importance of boundary conditions upon the Parker
instability, and demonstrate that a sufficiently strong magnetic field can
prevent the Parker instability from occurring. In addition, based on the
instability results, we further rigorously verify the Parker instability under
Schwarzschild's or Tserkovnikov's instability conditions in the sense of
Hadamard for a horizontally periodic domain.
