We employ a recently developed population-orbit superposition technique to simultaneously fit the stellar kinematic and age maps of 82 CALIFA spiral galaxies and obtain the ages of stars in different dynamical structures. We first evaluated the capabilities of this method on CALIFA-like mock data created from the Auriga simulations. The recovered mean ages of dynamically cold, warm, and hot components match the true values well, with an observational error of up to 20% in the mock age maps. For CALIFA spiral galaxies, we find that the stellar ages of the cold, warm, and hot components all increase with the stellar mass of the galaxies, from tcold ~ 2.2 Gyr, twarm ~ 2.3 Gyr, and thot ~ 2.6 Gyr for galaxies with stellar mass M* < 1010 M⊙, to tcold ~ 4.0 Gyr, twarm ~ 5.1 Gyr, and thot ~ 5.9 Gyr for galaxies with M* > 1011 M⊙. About 80% of the galaxies in our sample have thot > tcold, and the mean values of thot − tcold also increase with stellar mass, from 0.7−0.2+0.6 Gyr in low-mass galaxies (108.9 M⊙ < M* ≤ 1010.5 M⊙) to 1.7−0.2+0.7 Gyr in high-mass galaxies (1010.5 M⊙ < M* < 1011.3 M⊙). The stellar age is younger in disks than in bulges, on average. This suggests that either the disks formed later and/or that they experienced a more prolonged and extensive period of star formation. Lower-mass spiral galaxies have younger bulges and younger disks, while higher-mass spiral galaxies generally have older bulges, and their disks span a wide range of ages. This is consistent with the scenario in which the bulges in more massive spirals formed earlier than those in less massive spirals.