We introduce an exactly solvable model for glassy dynamics with many relaxational modes, each one characterized by a different relaxational time scale. Analytical solution of the aging dynamics at low temperatures shows that a nonequilibrium or effective temperature can be associated to each time scale or mode. The spectrum of effective temperatures shows two regions that are separated by an age-dependent boundary threshold. Region I is characterized by partially equilibrated modes that relax faster than the modes at the threshold boundary. Thermal fluctuations and time correlations for modes in region I show that those modes are in mutual thermal equilibrium at a unique age-dependent effective temperature s. In contrast, modes with relaxational time scales longer than that of modes at the threshold region II show diffusive properties and do not share the common temperature s. The shift of the threshold toward lower energy modes as the system ages, and the progressive shrinking of region II, determines how the full spectrum of modes equilibrates. As is usually done in experiments, we have defined a frequency-dependent effective temperature and we have found that all modes in region I are mutually equilibrated at the temperature s independently of the probing frequency. The present model aims to explain transport anomalies observed in supercooled liquids in terms of a collection of structurally disordered and cooperative rearranging mesoscopic regions.