Randomly censored survival data are frequently encountered in biomedical or reliability applications and clinical trial analyses. Testing the significance of statistical hypotheses is crucial in such analyses to get conclusive inference but the existing likelihood‐based tests, under a fully parametric model, are extremely nonrobust against outliers in the data. Although there exists a few robust estimators given randomly censored data, there is hardly any robust testing procedure available in the literature in this context. One of the major difficulties here is the construction of a suitable consistent estimator of the asymptotic variance of robust estimators, since the latter is a function of the unknown censoring distribution. In this article, we take the first step in this direction by proposing a consistent estimator of asymptotic variance of the M‐estimators based on randomly censored data without any assumption on the censoring scheme. We then describe and study a class of robust Wald‐type tests for parametric statistical hypothesis, both simple as well as composite, under such a set‐up. Robust tests for comparing two independent randomly censored samples and robust tests against one sided alternatives are also discussed. Their advantages and usefulness are demonstrated for the tests based on the minimum density power divergence estimators and illustrated with clinical trials and other medical data.