In this paper we consider a parameter estimation problem for delay differential equations that are typical in the bioscience. The problem is motivated by a model of the dynamics of the intracellular development cycle of Chlamydia Trachomatis and can be formulated as a inverse problem for a system of delay equations. We derive the continuous sensitivity equations and show that the corresponding initial value problem is well-posed. The sensitivity equations are used to compute gradients in the optimization loop for the parameter estimation problem. Numerical examples are given to illustrate the key ideas and we close with a note about the use of continuous sensitivity equations in the design of experiments.