Parylene is a widely used polymer possessing advantages such as simple and cheap production, possibility of fabrication on flexible substrates, transparency, and safety for the human body. Moreover, parylene can be used as an active layer of memristors-circuit design elements that are promising for the implementation of hardware neuromorphic systems. Recent studies show that memristors are not merely memory but also highly dynamical systems that can encode timing information. Here, a study of the switching kinetics and the timing second-order effects in memristors based on pristine and nanocomposite (with embedded silver nanoparticles) parylene is presented. The strong decrease in the resistive switching time and increase in the amplitude of the resistive state change after preliminary heating pulses are revealed. These effects are explained by the local heating of the parylene matrix by electric pulses, and the given explanation is supported by the numerical electrothermal model. Spike-timing-dependent plasticity with symmetrical nonoverlapping spikes is demonstrated. The obtained results indicate a possibility of the utilization of second-order effects in the development of the neuromorphic systems.