  Let $E \subset \mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We
prove that every bounded solution $u$ to $$\partial_tu- \Delta u=0, \quad
\text{in} \quad \mathbb R^{n+1}\setminus E$$ satisfies a Carleson measure
estimate condition. An important technical novelty of our work is that we
develop a corona domain approximation scheme for $E$ in terms of regular
Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic
version which is an improvement of the known results in that setting.
