This paper is concerned with the stability of a parallel flow of the compressible Navier-Stokes equation in a cylindrical domain. It is proved that the linearized semigroup around the parallel flow decays in the L²-norm as a one-dimensional heat semigroup when the Reynolds and Mach numbers are sufficiently small. The proof is given by combining the energy method of Iooss-Padula and a variant of the Matsumura-Nishida energy method.