This paper examines an event-triggered control design approach for discrete-timelinear parameter-varying (LPV) systems under control constraints. A parameter-dependentdynamic output-feedback controller with an event-triggering condition is designed to ensurethe regional asymptotic stability of the closed-loop system while minimizing a quadratic costfunction. Sufficient conditions are derived in terms of linear matrix inequalities (LMIs) thanksto the use of a parameter-dependent quadratic Lyapunov function. Also, convex optimizationschemes are proposed using these conditions to minimize an upper bound of a given cost functionor to maximize the size of the region of closed-loop stability. Examples illustrate the proposal