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