This paper shows that under an ISS-like condition on the zero-dynamics with time-delay, it is possible to achieve global adaptive regulation by dynamic partial state feedback for certain classes of time-delay cascade systems with nonlinear parameterization. The main contribution is the development of a delay-independent, partial state adaptive control scheme that employs a dynamic gain-based method, together with the adaptive feedback domination design and the technique of changing supply rate. By constructing suitable dynamic gain-based Lyapunov-Krasovskii functionals, we prove that all the states of the time-delay cascade system can be regulated to the origin while maintaining boundedness of the closed-loop system. (C) 2018 Elsevier Ltd. All rights reserved.