This article mainly solves the consensus issue of parabolic partial differential equation (PDE) agents with switching topology by output feedback. A novel edge-based adaptive control protocol is designed to reach consensus under the condition that the switching graphs are always connected at any switching instants. Different from the existing adaptive protocol associated with partial differential dynamics, the proposed adaptive observer-type law relies on the relative output information rather than relative state information. A proper Lyapunov functional is constructed and some important lemmas are used, then a sufficient condition is obtained for the consensus of parabolic PDE agents on switching graphs. Besides, a corollary about the distributed adaptive consensus of parabolic PDE agents on fixed undirected communication networks is given. Finally, the theoretical results are demonstrated by two numerical simulations.