We use recently derived ages for 61 Milky Way (MW) globular clusters (GCs) to show that their age-metallicity relation (AMR) can be divided into two distinct, parallel sequences at [Fe/H] ≳ -1.8. Approximately one-third of the clusters form an offset sequence that spans the full range in age (˜10.5-13 Gyr), but is more metal rich at a given age by ˜0.6 dex in [Fe/H]. All but one of the clusters in the offset sequence show orbital properties that are consistent with membership in the MW disc. They are not simply the most metal-rich GCs, which have long been known to have disc-like kinematics, but they are the most metal-rich clusters at all ages. The slope of the mass-metallicity relation (MMR) for galaxies implies that the offset in metallicity of the two branches of the AMR corresponds to a mass decrement of 2 dex, suggesting host galaxy masses of M_{*} ˜ 107-108 { M_{⊙}} for GCs that belong to the more metal poor AMR. We suggest that the metal-rich branch of the AMR consists of clusters that formed in situ in the disc, while the metal-poor GCs were formed in relatively low-mass (dwarf) galaxies and later accreted by the MW. The observed AMR of MW disc stars, and of the Large Magellanic Cloud, Small Magellanic Cloud and WLM dwarf galaxies, is shown to be consistent with this interpretation, and the relative distribution of implied progenitor masses for the halo GC clusters is in excellent agreement with the MW subhalo mass function predicted by simulations. A notable implication of the bifurcated AMR is that the identical mean ages and spread in ages, for the metal-rich and metal-poor GCs, are difficult to reconcile with an in situ formation for the latter population.