We present new near-infrared, JHKs, period-luminosity relations (PLRs) for RR Lyrae variables in the Messier 53 (M53 or NGC 5024) globular cluster. Multi-epoch JHKs observations, obtained with the WIRCam instrument on the 3.6 m Canada-France-Hawaii Telescope, are used for the first time to estimate precise mean magnitudes for 63 RR Lyrae stars in M53 including 29 fundamental-mode (RRab) and 34 first-overtone mode (RRc) variables. The JHKs-band PLRs for RR Lyrae stars are best constrained for RRab types with a minimal scatter of 22, 23, and 19 mmag, respectively. The combined sample of RR Lyrae is used to derive the Ks-band PLR, Ks = -2.303(0.063)logP + 15.212(0.016), exhibiting a 1σ dispersion of only 0.027 mag. Theoretical period-luminosity-metallicity (PLZ) relations are used to predict parallaxes for 400 Galactic RR Lyrae, resulting in a median parallax zero-point offset of -7 ± 3 μas in Gaia Early Data Release 3 (EDR3), which increases to 22 ± 2 μas if the parallax corrections are applied. We also estimate a robust distance modulus, μM53 = 16.403 ± 0.024 (statistical) ± 0.033 (systematic) mag, to M53 based on theoretical calibrations. Homogeneous and precise mean magnitudes for RR Lyrae in M53 together with similar literature data for M3, M4, M5, and ω Cen are used to empirically calibrate a new RR Lyrae PLZK relation, Ks = -0.848 (0.007) - 2.320(0.006)logP + 0.166(0.011) [Fe/H], anchored with Gaia EDR3 distances and theoretically predicted relations, and to simultaneously estimate precise RR Lyrae-based distances to these globular clusters.