  We prove the first positive results concerning boundary value problems in the
upper half-space of second order parabolic systems only assuming measurability
and some transversal regularity in the coefficients of the elliptic part. To do
so, we introduce and develop a first order strategy by means of a parabolic
Dirac operator at the boundary to obtain, in particular, Green's representation
for solutions in natural classes involving square functions and non-tangential
maximal functions, well-posedness results with data in $L^2$-Sobolev spaces
together with invertibility of layer potentials, and perturbation results. In
the way, we solve the Kato square root problem for parabolic operators with
coefficients of the elliptic part depending measurably on all variables. The
major new challenge, compared to the earlier results by one of us under time
and transversally independence of the coefficients, is to handle non-local
half-order derivatives in time which are unavoidable in our situation.
