The dielectric environment of atomically thin monolayer (ML) of semiconducting transition metal dichalcogenides affects both the electronic bandgap and the excitonic binding energy in the ML. We investigate the effect of the environment on the in-plane magnetic field brightening of neutral and charged dark exciton emissions in the WSe₂ ML. The monolayers placed in three dielectric environments are studied, in particular, the ML encapsulated in hexagonal BN (hBN) flakes, the ML deposited on a hBN layer, and the ML embedded between the hBN flake and SiO₂/Si substrate. We observe that the brightening rates of the neutral and charged dark excitons depend on the dielectric environment, which may be related to the variation of the level of carrier concentration in the ML. Moreover, the surrounding media, characterized by different dielectric constants, weakly influence the relative energies of the neutral and charged dark excitons in reference to the bright ones.