A variational calculation is performed to obtain the polaronic corrections to the ground and the first-excited-state energies of an electron in a parabolic quantum dot of a polar semiconductor for the entire range of the electron-phonon coupling constant and the confinement length. The number of virtual phonons, the size of the polaron and the polarization potential in the polaron ground state are also calculated. The theory is applied to both two- and three-dimensional GaAs quantum dots and it is shown that both the ground and the first-excited-state polaronic corrections in these dots can be considerably large if the dot sizes are of the order a few nanometres.