tokens: photometry for the faintest variable. Multi-epoch data were also used to investigate color–magnitude and Bailey diagrams for M53 RR Lyrae at NIR wavelengths. No significant overlap is seen in the color–magnitude plane between RRab and RRc stars, suggesting that horizontal branch evolution occurs toward the red edge of the instability strip (Arellano Ferro et al. 2012 ). RR Lyrae in the Bailey diagrams follow the expected locus of RRab in a typical OoII cluster. New NIR photometry for RR Lyrae in the metal-poor M53 GC provided an opportunity to constrain the metallicity dependence of their PLRs by means of a quantitative comparison with relatively metal-rich GCs. While single-epoch data for RR Lyrae are available for several GCs, for example from 2MASS, homogeneous JHK s -band time-series photometry is limited to only a few GCs. Time-series data are crucial to determine precise mean magnitudes and the PLRs for RR Lyrae with a dispersion of ∼0.04 mag, thus allowing a rigorous comparison of GCs with different mean metallicities. Combining precise K s -band PLRs in GCs covering a metallicity range of Δ[Fe/H] = 0.9 dex, we determined a metallicity dependence of the PLRs and simultaneously estimated distances to the GCs adopting different calibrations based on theoretical models and Gaia EDR3 data. The quantified metallicity dependence of ∼0.15–0.17 mag dex −1 is in good agreement with theoretical predictions for the relation from Marconi et al. ( 2015 ). When excluding the calibrator relations, the metallicity dependence of the PLRs in the GCs was found to be sensitive to the adopted spectroscopic metallicities, the RR Lyrae sample size, and the number of clusters, and can vary between 0.07 and 0.15 mag dex −1 . Therefore, homogeneous NIR time-series data in additional GCs, particularly in metal-rich ([Fe/H] > −1.0 dex) and more metal-poor ([Fe/H] −2.1 dex) GCs, is required to extend the metallicity baseline and the statistics of both the number of RR Lyrae and the GCs. Our analysis has shown that a robust constraint on the effects of metallicity on the PLRs and precise RR Lyrae distances to GCs can be obtained simultaneously, provided a statistically significant sample of GCs covering a wide range of mean metallicities is available. However, the calibration of the RR Lyrae PLZ relation in the Milky Way is still hampered by uncertainties and a possible zero-point offset in the parallaxes even with Gaia EDR3. Using theoretical PLZ relations to predict parallaxes for Galactic RR Lyrae, we found a small parallax zero-point offset of −7 ± 3 μ as in EDR3, which is significantly smaller than the value (−32 ± 4 μ as) obtained using the recipe of Lindegren et al. ( 2020a ). We note that our estimated parallax offsets are in agreement with the values provided by the corrections of Lindegren et al. ( 2020a ) for faint and distant RR Lyrae stars that have small parallaxes (0.5 mas). When Gaia EDR3 parallaxes were corrected using the formulae of Lindegren et
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