The Parker instability, which has been considered as a process governing the structure of the interstellar medium, is induced by the buoyancy of magnetic fields and cosmic rays. In previous studies, while the magnetic field has been fully incorporated in the context of isothermal magnetohydrodynamics, cosmic rays have normally been treated with the simplifying assumption of infinite diffusion along magnetic field lines but no diffusion across them. The cosmic-ray diffusion is, however, finite. In this work, we fully take into account the diffusion process of cosmic rays in a linear stability analysis of the Parker instability. Cosmic rays are described with the diffusion-convection equation. With realistic values of cosmic-ray diffusion coefficients expected in the interstellar medium, we show that the result of previous studies with the simplifying assumption about cosmic-ray diffusion applies well. The finiteness of the parallel diffusion decreases the growth rate of the Parker instability, while the relatively smaller perpendicular diffusion has no significant effect. We discuss the implication of our result on the role of the Parker instability in the interstellar medium.
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