Cox proportional hazard regression model is a popular tool to analyze the relationship between a censored lifetime variable with other relevant factors. The semi-parametric Cox model is widely used to study different types of data arising from applied disciplines like medical science, biology, reliability studies and many more. A fully parametric version of the Cox regression model, if properly specified, can yield more efficient parameter estimates leading to better insight generation. However, the existing maximum likelihood approach of generating inference under the fully parametric Cox regression model is highly non-robust against data-contamination which restricts its practical usage. In this paper we develop a robust estimation procedure for the parametric Cox regression model based on the minimum density power divergence approach. The proposed minimum density power divergence estimator is seen to produce highly robust estimates under data contamination with only a slight loss in efficiency under pure data. Further, they are always seen to generate more precise inference than the likelihood based estimates under the semi-parametric Cox models or their existing robust versions. We also sketch the derivation of the asymptotic properties of the proposed estimator using the martingale approach and justify their robustness theoretically through the influence function analysis. The practical applicability and usefulness of the proposal are illustrated through simulations and a real data example.