  The application of the parallax of pulsation (PoP) technique to determine
distances of pulsating stars implies the use of a scaling parameter, the
projection factor (p-factor), required to transform disc-integrated radial
velocities (RVs) into photospheric expansion velocities. The value of the
p-factor is poorly known and debated. Most PoP applications assume a constant
p-factor. However, it may actually depend on the physical parameters of each
star. We aim to calibrate p-factors for RR Lyrae stars (RRLs) and compare them
with classical Cepheids (CCs). Due to their higher surface gravity, RRLs have
more compact atmospheres, and provide a valuable comparison with their
supergiant siblings. We determined the p-factor of 17 RRLs using the SPIPS
code, constrained by Gaia DR3 parallaxes, photometry, and new RVs from the
OHP/SOPHIE spectrograph. We carefully examine the different steps of the PoP
technique, particularly the method to determine RV from spectra using the
classical cross-correlation function (CCF) approach. The method employed for RV
extraction from the CCF has a strong impact on the p-factor, of up to 10%.
However, this choice of method results in a global scaling of the p-factor,
marginally affecting the scatter within the sample for a given method. Over our
RRL sample, we find a mean value of $p = 1.248 \pm 0.022$ for RVs derived using
a Gaussian fit of the CCF. There is no evidence for a different value of the
p-factor of RRLs, although its distribution for RRLs appears significantly less
scattered than that for CCs. The p-factor does not appear to depend in a simple
way on fundamental stellar parameters. We argue that large-amplitude dynamical
phenomena occurring in the atmospheres of RRLs and CCs during their pulsation
affect the relative velocity of the spectral line-forming regions compared to
the velocity of the photosphere.
