  We give a simplified and direct proof of the Kato square root estimate for
parabolic operators with elliptic part in divergence form and coefficients
possibly depending on space and time in a merely measurable way. The argument
relies on the nowadays classical reduction to a quadratic estimate and a
Carleson-type inequality. The precise organization of the estimates is
different from earlier works. In particular, we succeed in separating space and
time variables almost completely despite the non-autonomous character of the
operator. Hence, we can allow for degenerate ellipticity dictated by a spatial
$A_2$-weight, which has not been treated before in this context.
