  The second data release of \it Gaia \rm revealed a parallax zero point offset
of $-0.029$~mas based on quasars. The value depended on the position on the
sky, and also likely on magnitude and colour. The offset and its dependence on
other parameters inhibited an improvement in the local distance scale using
e.g. the Cepheid and RR Lyrae period-luminosity relations. Analysis of the
recent \it Gaia \rm Early Data Release 3 (EDR3) reveals a mean parallax zero
point offset of $-0.021$~mas based on quasars. The \it Gaia \rm team addresses
the parallax zero point offset in detail and proposes a recipe to correct for
it, based on ecliptic latitude, $G$-band magnitude, and colour information.
This paper is a completely independent investigation into this issue focussing
on the spatial dependence of the correction based on quasars and the magnitude
dependence based on wide binaries. The spatial and magnitude corrections are
connected to each other in the overlap region between $17 < G < 19$. The
spatial correction is presented at several spatial resolutions based on the
HEALPix formalism. The colour dependence of the parallax offset is unclear and
in any case secondary to the spatial and magnitude dependence. The spatial and
magnitude corrections are applied to two samples of brighter sources, namely a
sample of $\sim$100 stars with independent trigonometric parallax measurements
from \it HST \rm data, and a sample of 75 classical cepheids using photometric
parallaxes. The mean offset between the observed GEDR3 parallax and the
independent trigonometric parallax (excluding outliers) is about $-39$~\muas,
and after applying the correction it is consistent with being zero. For the
classical cepheid sample it is suggested that the photometric parallaxes may be
underestimated by about 5\%.
