The ground-state energy, addition energies, and optical absorption spectra are derived for interacting polarons in parabolic quantum dots in three and two dimensions. A path integral formalism for identical particles is used in order to take into account the fermion statistics. The approach is applied to both closed-shell and open-shell systems of interacting polarons. Using a generalization of the Jensen-Feynman variational principle, the ground-state energy of a confined N-polaron system is analyzed as a function of N and of the electron-phonon coupling constant α. In contrast to few-electron systems without the electron-phonon interaction, three types of spin polarization are possible for the ground state of the few-polaron systems: (i) a spin-polarized state, (ii) a state where the spin is determined by Hund's rule, and (iii) a state with the minimal possible spin. A transition from a state fulfilling Hund's rule to a spin-polarized state occurs when the electron density is decreased. In the strong-coupling limit, the system of interacting polarons turns into a state with the minimal possible spin. These transitions should be experimentally observable in the optical absorption spectra of quantum dots.