We consider linear nonautonomous second order parabolic
equations on bounded domains subject to  Dirichlet boundary
condition. Under  mild regularity assumptions on the
coefficients and the domain, we establish the existence
of a principal Floquet bundle  exponentially separated from a
complementary invariant bundle. Our main theorem extends
in a natural way standard results on  principal eigenvalues and
eigenfunctions of elliptic and time-periodic parabolic equations.
Similar theorems  were earlier  available only for smooth domains
and coefficients. As a corollary of our main result, we obtain
the uniqueness of positive entire solutions of the equations
in question.