  As primary anchors of the distance scale, Cepheid stars play a crucial role
in our understanding of the distance scale of the Universe because of their
period-luminosity relation. Determining precise and consistent parameters
(radius, temperature, color excess, and projection factor) of Cepheid pulsating
stars is therefore very important. With the high-precision parallaxes delivered
by the early third Gaia data release, we aim to derive various parameters of
Cepheid stars in order to calibrate the period-luminosity and period-radius
relations and to investigate the relation of period to p-factor. We applied an
implementation of the parallax-of-pulsation method through the algorithm called
Spectro-Photo-Interferometry of Pulsating Stars, which combines all types of
available data for a variable star in a global modeling of its pulsation. We
present the SPIPS modeling of a sample of 63 Galactic Cepheids. Adopting Gaia
EDR3 parallaxes as an input associated with the best available dataset, we
derive consistent values of parameters for these stars such as the radius,
multiband apparent magnitudes, effective temperatures, color excesses, period
changes, Fourier parameters, and the projection factor. We then derive new
calibrations of the period-luminosity and period-radius relations. After
investigating the dependences of the p-factor on the parameters of the stars,
we find a high dispersion of its values and no evidence of its correlation with
the period or with any other parameters. Statistically, the p-factor has an
average value of p=1.26$\pm$0.07, but with an unsatisfactory agreement. In
absence of any clear correlation between the p-factor and other quantities, the
best agreement is obtained under the assumption that the p-factor can take any
value in a band with a width of 0.15. This result highlights the need for a
further examination of the physics behind the p-factor.
