  We study parabolic chord arc domains, introduced by Hofmann, Lewis and
Nystr\"om, and prove a free boundary regularity result below the continuous
threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc
domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing
chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This
generalizes, to the parabolic setting, a result of Kenig and Toro and answers
in the affirmative a question left open in the aforementioned paper of Hofmann
et al. A key step in this proof is a classification of "flat" blowups for the
parabolic problem.
