Optically resonant nanoparticles decorated with donor-acceptor molecular pairs have been attracting attention for applications as nanoprobes in bioimaging and biosensing. We produced composite nanoparticles composed of donor-acceptor molecular pairs and silicon nanospheres (Si NSs) with diameters of 100-200 nm exhibiting Mie resonances in the visible range and studied the effect of Mie resonances on their photoluminescence properties. We showed that the photoluminescence spectra are strongly modified by Mie resonances and the spectral shape is controlled in a wide range by the Si NS size; by controlling the size, we can achieve the photoluminescence maximum from that of a donor molecule to that of an acceptor molecule almost continuously. From the photoluminescence decay properties in combination with theoretical calculations, we showed that the observed strong modification of the spectral shape is mainly due to the Purcell effect on donor and acceptor molecules, and the effect of Mie resonances on the Förster resonance energy transfer (FRET) rate is relatively small. We also showed that because of the large Purcell effect and the small FRET rate enhancement, Mie resonances decrease the FRET efficiency.