In Parkinson's disease (PD), there is evidence that α-synuclein (αSN) aggregation is coupled to dysfunctional or overburdened protein quality control systems, in particular the ubiquitin–proteasome system. Here, we develop a simple dynamical model for the on-going conflict between αSN aggregation and the maintenance of a functional proteasome in the healthy cell, based on the premise that proteasomal activity can be titrated out by mature αSN fibrils and their protofilament precursors. In the presence of excess proteasomes the cell easily maintains homeostasis. However, when the ratio between the available proteasome and the αSN protofilaments is reduced below a threshold level, we predict a collapse of homeostasis and onset of oscillations in the proteasome concentration. Depleted proteasome opens for accumulation of oligomers. Our analysis suggests that the onset of PD is associated with a proteasome population that becomes occupied in periodic degradation of aggregates. This behavior is found to be the general state of a proteasome/chaperone system under pressure, and suggests new interpretations of other diseases where protein aggregation could stress elements of the protein quality control system.