Early detection is vital for future neuroprotective treatments of Parkinson's disease (PD). Resting state electroencephalographic (EEG) recording has shown potential as a cost-effective means to aid in detection of neurological disorders such as PD. In this study, we investigated how the number and placement of electrodes affects classifying PD patients and healthy controls using machine learning based on EEG sample entropy. We used a custom budget-based search algorithm for selecting optimized sets of channels for classification, and iterated over variable channel budgets to investigate changes in classification performance. Our data consisted of 60-channel EEG collected at three different recording sites, each of which included observations collected both eyes open (total N = 178) and eyes closed (total N = 131). Our results with the data recorded eyes open demonstrated reasonable classification performance (ACC = .76; AUC = .76) with only 5 channels placed far away from each other, the selected regions including right-frontal, left-temporal and midline-occipital sites. Comparison to randomly selected subsets of channels indicated improved classifier performance only with relatively small channel-budgets. The results with the data recorded eyes closed demonstrated consistently worse classification performance (when compared to eyes open data), and classifier performance improved more steadily as a function of number of channels. In summary, our results suggest that a small subset of electrodes of an EEG recording can suffice for detecting PD with a classification performance on par with a full set of electrodes. Furthermore our results demonstrate that separately collected EEG data sets can be used for pooled machine learning based PD detection with reasonable classification performance.