Parkinson's disease (PD) is a neurodegenerative disorder that negatively affects millions of people. PD is usually diagnosed by a series of empirical tests and sometimes by invasive methods. Classifying People with Parkinsonism (PWP) from healthy people using speech signals may lead to innovative, noninvasive PD diagnosis. In this study, we developed a machine learning system to classify PWP using their speech signals. In the system, four feature selection algorithms, six classifiers, and two validation methods were employed for accurate classification of PWP. The system calculated the accuracy, sensitivity, specificity, and Matthews correlation coefficient of the results. Additionally, the execution times of the algorithms were computed. All utilized algorithms, classifiers, validation methods, and evaluation metrics are briefly reviewed in the article. The main innovative part of this study is developing a comprehensive machine learning system for classifying PWP and testing it on a PD dataset, which consisted of multiple types of speech signals. Applying feature selection methods greatly increased the accuracy of classification. The most significant and discriminative features of speech signals were obtained and explained with a medical background. The importance of the selected features is also evaluated from the medical perspective.