Optical amplifiers in all ranges of the electromagnetic spectrum exhibit an essential characteristic, namely the input signal during the propagation in the amplifier medium is multiplied by the avalanche effect of the stimulated emission to produce exponential growth. We perform a theoretical study motivated and supported by experimental data on a He gas amplifier driven by intense 30-fs-long laser pulses and seeded with attosecond pulse trains generated in a separated Ne gas jet. We demonstrate that the strong-field theory in the frame of high harmonic generation fully supports the appearance of the avalanche effect in the amplification of extreme ultraviolet attosecond pulse trains. We theoretically separate and identify different physical processes taking part in the interaction and we demonstrate that X-ray parametric amplification dominates over others. In particular, we identify strong-field mediated intrapulse X-ray parametric processes as decisive for amplification at the single-atom level. We confirm that the amplification takes place at photon energies where the amplifier is seeded and when the seed pulses are perfectly synchronized with the driving strong field in the amplifier. Furthermore, propagation effects, phase matching and seed synchronization can be exploited to tune the amplified spectral range within the seed bandwidth.