Cepheid metallicity in the Leavitt law (C-MetaLL) survey - III. Simultaneous derivation of the Gaia parallax offset and period-luminosity-metallicity coefficients

Classical Cepheids (DCEPs) are the most important standard candles in the extra-galactic distance scale thanks to the period-luminosity ($\rm PL$), period-luminosity-colour ($\rm PLC$), and period-Wesenheit ($\rm PW$) relations that hold for these objects. The advent of the Gaia mission, and in particular the Early Data Release 3 (EDR3), provided accurate parallaxes to calibrate these relations. In order to fully exploit Gaia measurements, the zero point (ZP) of Gaia parallaxes should be determined with an accuracy of a few $\rm \mu as$. The individual ZP corrections provided by the Gaia team depend on the magnitude and the position on the sky of the target. In this paper, we use an implicit method that relies on the Cepheid $\rm PL$ and $\rm PW$ relations to evaluate the ensemble Gaia parallax zero point. The best inferred estimation of the offset value needed to additionally correct (after the Gaia team correction) the Gaia parallaxes of the present DCEP sample amounts to $\rm -22\pm 4\, \mu as$. This value is in agreement with the most recent literature values and confirms that the correction proposed by the Gaia team overcorrected the parallaxes. As a further application of our results, we derive an estimate of the Large Magellanic Cloud distance ($\rm \mu _0=18.49\pm 0.06\, mag$) in very good agreement with the currently accepted value obtained through geometric methods.