  Parallaxes measured by the Gaia mission have huge significance for astronomy,
but parallaxes in Gaia DR2 are known to have systematic errors that depend on
the source position and other quantities. We use the abundant information in
faint Milky Way stars, along with the GOG simulation of the Gaia catalog, to
probe the spatial dependence of Gaia DR2 parallax systematic errors in an
empirical way. The parallax signal, concentrated in thick disk turnoff stars
with magnitude G ~ 17, is sufficient to construct maps of the parallax
systematic error over the majority of the sky. These maps show a locally
regular "waffle pattern" on ~1 degree scales following Gaia scan directions,
stronger linear "scar" features, and coherent variations on larger scales. The
parallax bias maps also retain traces of astrophysical effects such as dust
clouds. The waffle pattern, known from earlier maps of the Magellanic Clouds,
extends over the entire sky; its local rms amplitude averages 15 microarcsec
and varies by about a factor of two. The strength of this pattern increases by
a factor ~6 from magnitude G = 13 to G = 20. Correlations with parallaxes of
quasars and of stars with independent distance estimates support our bias
estimates. Using similar methods, we map systematic errors in the proper motion
and examine the relationship with the parallax systematics. We provide a code
package to access and query our bias maps. Similar tests on the general stellar
population should be useful in quantifying systematic errors in future Gaia
releases.
