All-dielectric "magnetic light" nanophotonics based on high refractive index nanoparticles allows controlling magnetic component of light at nanoscale without having high dissipative losses. The artificial magnetic optical response of such nanoparticles originates from circular displacement currents excited inside those structures and strongly depends on geometry and dispersion of optical materials. Here an approach for enhancing of magnetic response via resonant bianisotropy effect is proposed and analyzed. The key mechanism of enhancement is based on electric-magnetic interaction between two electrically and magnetically resonant nanoparticles of all-dielectric dimer. It was shown that proper geometrical arrangement of the dimer in respect to the incident illumination direction allows flexible control over all vectorial components of the magnetic moment, tailoring the latter in the dynamical range of 100% and delivering enhancement up to 36% relative to performances of standalone spherical particles. The proposed approach provides pathways for designs of all-dielectric metamaterials and metasurfaces with strong magnetic responses.