For the solution of the generalized complex non-Hermitian eigenvalue problems $Ax=\lambda Bx$ occurring in the spectral study of linearized resistive magnetohydrodynamics (MHD) a new parallel solver based on the recently developed Jacobi-Davidson~\cite{Sleijpen96a} method has been developed. A brief presentation of the implementation of the solver is given here. The new solver is very well suited for the computation of some selected interior eigenvalues related to the resistive Alfv\'{e}n wave spectrum and is well parallelizable. All features of the spectrum are easily and accurately computed with only a few target shifts.