This paper deals with new finite-time estimation algorithms for Linear Parameter Varying (LPV) discrete-time systems and their application to output feedback stabilization. Two exact finite-time estimation schemes are proposed. The first one provides a direct and explicit estimation algorithm based on the use of delayed outputs, while the second one uses two combined asymptotic observers, connected by a condition of invertibility of a certain time-varying matrix, to recover in a finite-time the solution of the LPV system. Furthermore, a stabilization strategy is proposed as an extension. This strategy, called Two Connected Observers Feedback (2-COF) stabilization method, is based on the use of the two combined observers based estimation algorithm.