InP-based colloidal quantum dots are promising for optoelectronic devices such as light-emitting diodes and lasers. Understanding and optimizing their emission process is of scientific interest and essential for large-scale applications. Here we present a study of the exciton recombination dynamics in InP QDs with various shells: ZnS, ZnSe, and (Zn,Cd)Se with different amounts of Cd (5, 9, 12%). Phonon energies extracted from Raman spectroscopy measurements at cryogenic temperatures (4-5 K) are compared with exciton emission peaks observed in fluorescence line narrowing spectra. This allowed us to determine the position of both the bright F=±1 state and the lowest dark F=±2 state. We could identify the phonon modes involved in the radiative recombination of the dark state and found that acoustic and optical phonons of both the core and the shell are involved in this process. The Cd content in the shell increases electron wave-function delocalization, and thereby enhances the exciton-phonon coupling through the Fröhlich interaction.