Abstract: We perform numerical magnetohydrodynamic (MHD) simulations of the gravitational collapse and fragmentation of a cylindrical molecular cloud with the help of the FLASH code. The cloud collapses rapidly along its radius without any signs of fragmentation in the simulations without magnetic field. The radial collapse of the cloud is stopped by the magnetic pressure gradient in the simulations with parallel magnetic field. Cores with high density form at the cloud’s ends during further evolution. The core densities are and cm–3 in the cases with initial magnetic field strengths and G, respectively. The cores move toward the cloud’s center with supersonic speeds and km/s. The sizes of the cores along the filaments radius and filament’s main axis are pc and pc, pc and pc, respectively. The masses of the cores increase during the filament evolution and lie in range of. According to our results, the cores observed at the edges of molecular filaments can be a result of the filament evolution with parallel magnetic field. © Pleiades Publishing, Ltd. 2024. ISSN 1063-7729, Astronomy Reports, 2024, Vol. 68, No. 1, pp. 60–66. Pleiades Publishing, Ltd., 2024.