Necessary and sufficient conditions are established for the parameter redundancy of a wide class of nonlinear models for data distributed according to the exponential family. The likelihood surfaces for parameter-redundant models possess completely flat ridges. Whether a model is parameter redundant can be established by checking the rank of a derivative matrix, using a symbolic algebra package. A feature of contingency table applications is the need to extend conclusions from particular to general dimensions. We meet this via an extension theorem. Examples are given from the area of animal survival estimation using mark-recapture/recovery data.