We consider the Dirichlet problem for a class of semilinear parabolic equation on a bounded domain which is reflectionally symmetric about a hyperplane H. The equation consists of a symmetric time-autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity. In our first theorem, we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem. The novelty of this result is that the  solutions considered are not assumed uniformly positive, which  prevents one from applying common techniques based on the method of moving hyperplanes. In our second main theorem, we classify the positive entire solutions of the unperturbed time-autonomous problem. In particular, we characterize all entire solutions, which are not symmetrically decreasing in the direction orthogonal to H, as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set.