We consider fully nonlinear parabolic equations on nonsmooth bounded domains.
Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that
each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared
with previous results of this type, we do not assume certain crucial hypotheses, such as uniform
(with respect to time) positivity of the solution or regularity of the nonlinearity in time.  Our method
is based on estimates of solutions of linear parabolic problems, in particular on a theorem on asymptotic
positivity of such solutions.