If Ω is a Lip(1,1/2) domain, μ a doubling measure on ${\partial }_p\Omega ,\partial /\partial t-L_i$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ${\omega }_0$, ${\omega }_1$ have the property that ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1$ is absolutely continuous with respect to ${\omega }_0$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_1$ and $L_0$. Also ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1\in B^q\left(\mu \right)$ whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures extended...