We analyze quantitatively the oxidative degradation of a sample in a parallel-disk rheometer, as oxygen diffuses inward, radially, from the free boundary. We examine rheometer error mitigation by means of nitrogen blanketing, and also, of parallel-disk partitioning. We arrive at exact analytical expressions for the oxygen concentration and, thus, for the degradation rate. We then integrate this rate over time to get the amount of oxygen reacted as a function of radial position and time in the degrading sample. To illustrate the usefulness of our analytical expressions, we provide two worked examples investigating the effect of nitrogen blanketing and parallel-disk partitioning. We find that, though nitrogen blanketing always produces less degradation, its benefits are limited for short times. Additionally, parallel-disk partitioning provides a simpler solution and allows samples to be run for longer times without degradation compromising measurement, even in samples initially saturated with oxygen. We also consider the effect of antioxidants. We also consider an important special case, without chemical reaction, where the sample dries by evaporation from its free surface. We close by comparing the roles played by polymer degradation in parallel-disk flow versus cone-plate flow.