Record linkage is the act of bringing together records that are believed to belong to the same unit (e.g., person or business) from two or more files. Record linkage is not an error-free process and can lead to linking a pair of records that do not belong to the same unit. This occurs because linking fields on the files, which ideally would uniquely identify each unit, are often imperfect. There has been an explosion of record linkage applications, particularly involving government agencies and in the field of health, yet there has been little work on making correct inference using such linked files. Naively treating a linked file as if it were linked without errors can lead to biased inferences. This article develops a method of making inferences for cross tabulated variables when record linkage is not an error-free process. In particular, it develops a parametric bootstrap approach to estimation which can accommodate the sophisticated probabilistic record linkage techniques that are widely used in practice (e.g., 1-1 linkage). The article demonstrates the effectiveness of this method in a simulation and in a real application.