Stellar ages are an important parameter in studies of the chemical evolution of the Galaxy. To better estimate these ages, various methods complementary to the conventional isochrone fitting method have been implemented in the past decade. Several recent studies have established the existence of a relationship between chemical clocks and stellar ages. The [Y/Mg] clock is a promising technique, but there are still several open questions, such as its validity for metal-poor stars and differences between the thin and thick disk populations. Our aim is to study the relationship between the [Y/Mg] chemical clock and stellar ages for a sample of solar-type disk stars and to provide the empirical dating relation(s) for the stellar age determination from their precise chemical abundances. We also studied the effect of metallicity and populations on this chemical clock. We derived precise stellar atmospheric parameters as well as the elemental abundances of Mg and Y through line-by-line differential spectroscopic analysis for a sample of 48 metal-poor solar-type stars based on high-quality, high-resolution ESO/HARPS spectra. From high-precision Gaia astrometric data, stellar masses and ages were estimated through isochrone fitting using Yonsei-Yale isochrones. A joint analysis of our sample, together with a sample of 185 solar twins and analogues from our previous works, was performed to calibrate the [Y/Mg] chemical clock in the Galactic disk for -0.71<=[Fe/H]<+0.34. Open clusters and stars with asteroseismic ages were used to validate our relations. Two different populations are clearly seen in the [Mg/Fe]-[Fe/H] plane: the thick and thin disks. Thick disk stars show an age-metallicity relation, whereas the thin disk shows a flatter age-metallicity distribution. We find a strong, metallicity-dependent anti-correlation between the [Y/Mg] ratio and the stellar ages of our sample. For the first time in the literature, we report similar correlations for thin and thick disk stars. We find that the [Y/Mg] relation(s) found here for solar-type stars in a wide metallicity range are compatible with those found for solar twins in the literature. Our relation provides high accuracy and precision (0.45 and 0.99 Gyr, respectively) comparable with the best accuracy achieved for solar twins to date.