In regular state stabilization of a class of linear parabolic systems, the number of the sensors wk and the actuators hk are required to be greater than or equal to the maximum of the multiplicities of the unstable eigenvalues. In this paper, we ask what control theoretic results we can establish with smaller numbers of wk and hk. The enhancement of stability of output or the output stabilization, a concept weaker than regular state stabilization, gives an answer to the question. Conditions of a fairly different nature than those for regular state stabilization appear; these are the rank conditions of the products of the observability and controllability matrices. The result is applied to a parabolic system with boundary observation/control via two different approaches.