Relational arrays represent interactions or associations between pairs of actors, often over time or space. We focus on the case where the elements of a relational array are modeled as a linear function of observable covariates. Due to the inherent dependencies among relations involving the same individuals, standard regression methods for quantifying uncertainty for independent data are invalid. Furthermore, existing estimators that recognize relational dependence rely on estimating complex structure with very limited data. By assuming the data are partially exchangeable, we derive parsimonious standard error estimators with substantially better performance than existing estimators. This exchangeability assumption is pervasive in network and array models in the statistics literature, but not typically considered when adjusting for dependence in network regressions. We demonstrate the improvements in inference that result from using our proposed estimator through simulation and a dataset involving international trade.