We present a detailed theoretical description of laser driven x-ray parametric amplification, which has been experimentally demonstrated by Seres et al. [Nature Phys. 6, 455 (2010)] together with a supporting basic model. The process is based on the parametric interaction of an x-ray photon with a laser accelerated electron in a Coulomb field. With the extended model we are able to estimate the gain cross section also for a finite energy distribution of the interacting electrons as well as to consider dephasing between the electrons and the x-ray field. The improved model is capable of describing the recent experimental findings much more accurately.