This work focuses on predictive output feedback control of linear parabolic partial differential equation (PDE) systems with state and control constraints. Under the assumption that a finite, yet sufficiently large, number of output measurements is available, two predictive output feedback controllers are constructed and sufficient conditions are derived under which these controllers enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system, respectively. The controllers are applied to an example of a linear parabolic PDE with Dirichlet boundary conditions subject to state and control constraints, and the numerical simulations demonstrate their ability to enforce closed-loop system stability and constraint satisfaction.