We introduce an automated method for parameterized verification of fault-tolerant distribed algorithms. It rests on a novel parametric interval abstraction (PIA) technique, which works for systems with multiple parameters, for instance, where n and t are parameters describing the system size and the bound on the number of faulty processes, respectively. The PIA technique allows to map typical threshold-range intervals like [1,t+1) and [t+1,n-t) to values from a finite abstract domain. Applying PIA to both the local states of the processes and the global system state, the parameterized verification problem can be reduced to finite-state model checking. We demonstrate the practical feasibility of our method by verifying several variants of the well-known consistent broadcasting algorithm by Srikanth and Toueg for different fault models. To the best of our knowledge, this is the first successful automated parameterized verification of a Byzantine fault-tolerant distributed algorithm for message-passing systems.