In this paper, the resistive switching and neuromorphic behaviour of memristive devices based on parylene, a polymer both low-cost and safe for the human body, is comprehensively studied. The Metal/Parylene/ITO sandwich structures were prepared by means of the standard gas phase surface polymerization method with different top active metal electrodes (Ag, Al, Cu or Ti of ~500 nm thickness). These organic memristive devices exhibit excellent performance: low switching voltage (down to 1 V), large OFF/ON resistance ratio (up to 104), retention (≥104 s) and high multilevel resistance switching (at least 16 stable resistive states in the case of Cu electrodes). We have experimentally shown that parylene-based memristive elements can be trained by a biologically inspired spike-timing-dependent plasticity (STDP) mechanism. The obtained results have been used to implement a simple neuromorphic network model of classical conditioning. The described advantages allow considering parylene-based organic memristors as prospective devices for hardware realization of spiking artificial neuron networks capable of supervised and unsupervised learning and suitable for biomedical applications.