We introduce an automated parameterized verification method for fault-tolerant distributed algorithms (FTDA). FTDAs are parameterized by both the number of processes and the assumed maximum number of faults. At the center of our technique is a parametric interval abstraction (PIA) where the interval boundaries are arithmetic expressions over parameters. Using PIA for both data abstraction and a new form of counter abstraction, we reduce the parameterized problem to finite-state model checking. We demonstrate the practical feasibility of our method by verifying safety and liveness of several fault-tolerant broadcasting algorithms, and finding counter examples in the case where there are more faults than the FTDA was designed for.