  We present a detailed study of the growth of the Parker instability in a
differentially rotating disk embedded in an azimuthal equilibrium magnetic
field, such as the interstellar gas or an accretion disk. Basic properties of
the instability without shear are first recalled. Differential rotation is
modeled in the shearing sheet approximation, classical in the theory of spiral
density waves. The action of differential rotation is reduced to two different
effects, (i) a linear time-dependence of the radial wavenumber, and (ii) a
radial differential force. We present both exact numerical solutions, and
approximate analytical ones based on the WKB approximation in the limit of weak
differential rotation. Most important are (i) a transient natural stabilization
of the Parker mode due to the radial differential force (ii) the generation of
magnetosonic and Alfvenic waves, and (iii) in a certain parameter range a
possible ``turn-over'' of the perturbation whereby, quite surprisingly, matter
which had started being elevated by the instability may end up dropping towards
the disk midplane. A simplified model shows the possible observable effects of
this turn-over.
