In this talk, I examine the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Such a differential equation can be written as the L^2 gradient of an appropriate functional, so one might expect to be able to do Morse theory on it. However, the structure of the equilibrium solutions is fairly delicate, and exhibits some surprising bifurcations. The exact nature of this bifurcation behavior leads to a demonstration that the equilibria are degenerate critical points in the sense of Morse. Instead of following the usual Morse-theoretic route, we instead focus on the space of heteroclinic orbits (following Floer) to find that the space of heteroclinic orbits has a cell complex structure, which is finite dimensional when the number of equilibria is finite.