In this paper, we describe the features of oscillations in adiabatic oscillators when the delay is introduced into the equation. We give a short description of the method of asymptotic integration of one class of linear delay differential systems in the neighborhood of infinity. This method is based on the idea of transforming the initial system in order to reduce it to the system that is close in some sense to the system of ordinary differential equations. When applying this method, we need to extend the phase space of the initial system. The averaging changes of variables are also used to simplify the procedure of constructing the asymptotic formulas. Finally, we apply the functional differential analog of the Levinson theorem. We use this method to get the asymptotic formulas for adiabatic oscillators with delay under a monotonely and also oscillatory tending to zero perturbations. In conclusion, we study the transformation of the parametric resonance zone of one adiabatic oscillator when the delay is varied.