What's the relation between the parallax distance and the luminosity distance?

Question

I have read that Riess and his team are able to measure $H_0$ from supernovae calibrated using Cepheid in a model independent way.

from what I have gathered they find the absolute luminosity of Cepheid $M_c$ with the parallax method and a bunch of other geometric methods then, once they have $ M_c$, they use the luminosity redshift relation of SN Ia to find $M_{sn}$.

my question is:

once they have the parallax distance of Cepheid, how do they find their distance moduli without imposing a model? in other words, how do they convert the parallax distance in the luminosity distance without assuming a model? otherwise I don't understand how can they find their absolute luminosity or the distance moduli needed to find $M_c$

p.s. i had initially posted in physics.stackexchange but someone told me to post here

Answer 1

The distinction between different distance measures in cosmology (in this context, the luminosity distance and the parallax distance) only becomes significant over cosmological distances, i.e., when the redshift of the objects begins to approach $\mathcal{O}(1)$.

Maybe one day we will be able to observe parallax for cosmologically distant objects. Then it will be important to consider how parallax distances compare to luminosity distances. That day is not today, however. We only measure parallax distances for objects in our own galaxy. The cosmological model has no relevance in this context.