  We study boundary regularity for the normalized $p\mspace{1mu}$-parabolic
equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math.
J. 20 (1970), 683-693) showed that the so-called tusk condition guarantees
regularity for the heat equation. We generalize this result to the normalized
$p\mspace{1mu}$-parabolic equation, and also obtain H\"older continuity. The
tusk condition is a parabolic version of the exterior cone condition. We also
obtain a sharp Petrovski criterion for the regularity of the latest moment of a
domain. This criterion implies that the regularity of a boundary point is
affected if one side of the equation is multiplied by a constant.
