We perform three-dimensional numerical MHD simulations of the gravitational fragmentation of the molecular filaments with a large-scale parallel magnetic field. We use the numerical code FLASH. The filaments have typical length of 2 pc and width of 0.1 pc and various magnetic field strengths. We consider two fragmentation mechanisms: the instability of small perturbations of a filament surface and the gravitational instability of magnetosonic waves propagating along a filament. The simulations show that gravitationally-bound cores form when the perturbation wavelength is equal to or greater than critical wavelength. The masses of the cores increase with the perturbation wavelength and range from 0.3 to 3 M ⊙ . The number of cores depends on the perturbation wavelength and ranges from 4 to 13. The cores have almost spherical form just after their formation. During further evolution, the cores become more elongated along the filament axis due to the asymmetry of the cylindrical collapse. In addition to the perturbations growth within the filament, gravitational focusing leads to the formation of the cores at the filament edges. These cores go through the isothermal stage of the collapse faster than the fragments inside of the filament. This means that the stars formed at the filament edges will be older than the stars inside the filament.