Core/shell CdSe/(Cd,Mn)S colloidal nanoplatelets containing magnetic Mn2+ ions are investigated by the optically detected magnetic resonance technique, combining 60 GHz microwave excitation and photoluminescence detection. Resonant heating of the Mn spin system is observed. We identify two mechanisms of optical detection, via variation of either the photoluminescence polarization or its intensity in an external magnetic field. The spin-lattice relaxation dynamics of the Mn spin system is measured and used for evaluation of the Mn concentration. In CdSe/(Cd,Zn,Mn)S nanoplatelets the addition of Zn in the shells significantly broadens the magnetic resonance, evidencing local strain.