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Classical double-mode pulsators (RR Lyrae stars and delta Cepheids) areimportant for their simultaneous pulsation in low-order radial modes. Thisenables us to put stringent constraints on their physical parameters.  We use 30 bright galactic double-mode RR~Lyrae (RRd) stars to estimate theirluminosities and compare them with those derived from the parallaxes of therecent data release (EDR3) of the Gaia survey.  We employ pulsation and evolutionary models, together with observationallydetermined effective temperatures to derive the basic stellar parameters.  Excluding 6 outlying stars (e.g., with blending issues) the RRd and Gaialuminosities correlate well. With the adopted temperature zero point from oneof the works based on the infrared flux method, we find it necessary toincrease the Gaia parallaxes by 0.02 mas to bring the RRd and Gaia luminositiesinto agreement. This value is consonant with those derived from studies onbinary stars in the context of Gaia. We examine also the resultingperiod-luminosity-metallicity (PLZ) relation in the 2MASS K band as followsfrom the RRd parameters. This leads to the verification of two independentlyderived other PLZs. No significant zero point differences are found.Furthermore, the predicted K absolute magnitudes agree withinsigma=0.005-0.01mag.

Probing galactic double-mode RR Lyrae stars against Gaia EDR3