In this paper, we study H-infinity control of linear parabolic systems with unbounded output operators. Our aim is to construct a finite-dimensional stabilizing controller for the linear parabolic system that makes the H-infinity norm of the closed-loop transfer function less than a given positive number delta. To this end, we first derive a finite-dimensional reduced-order system for the linear parabolic system. Then, for the reduced-order model, we construct a stabilizing controller that makes the H-infinity norm of the closed-loop transfer function less than another positive number. It is proved that the finite-dimensional controller together with a residual mode filter plays a role of a finite-dimensional stabilizing controller that makes the H-infinity norm of the closed-loop transfer function less than delta for the original linear parabolic system, if the order of residual mode filter is chosen sufficiently large.