We use a sample of 809 photometrically classified Type Ia supernovae (SNe Ia) discovered by the Dark Energy Survey (DES) along with 40 415 field galaxies to calculate the rate of SNe Ia per galaxy in the redshift range 0.2 < z < 0.6. We recover the known correlation between SN Ia rate and galaxy stellar mass across a broad range of scales 8.5 ≤ log (M*/M⊙) ≤ 11.25. We find that the SN Ia rate increases with stellar mass as a power law with index 0.63 ± 0.02, which is consistent with the previous work. We use an empirical model of stellar mass assembly to estimate the average star formation histories (SFHs) of galaxies across the stellar mass range of our measurement. Combining the modelled SFHs with the SN Ia rates to estimate constraints on the SN Ia delay time distribution (DTD), we find that the data are fit well by a power-law DTD with slope index β = -1.13 ± 0.05 and normalization A = 2.11 ± 0.05 × 10-13 SNe M⊙-1 yr-1, which corresponds to an overall SN Ia production efficiency $N_{\mathrm{Ia}}/M_* = 0.9~_{-0.7}^{+4.0} \times 10^{-3}~\mathrm{SNe}~\mathrm{M}_{\odot }^{-1}$. Upon splitting the SN sample by properties of the light curves, we find a strong dependence on DTD slope with the SN decline rate, with slower-declining SNe exhibiting a steeper DTD slope. We interpret this as a result of a relationship between intrinsic luminosity and progenitor age, and explore the implications of the result in the context of SN Ia progenitors.