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Parkinson's disease dysgraphia (PDYS), one of the earliest signs of Parkinson's disease (PD), has been researched as a promising biomarker of PD and as the target of a noninvasive and inexpensive approach to monitoring the progress of the disease. However, although several approaches to supportive PDYS diagnosis have been proposed (mainly based on handcrafted features (HF) extracted from online handwriting or the utilization of deep neural networks), it remains unclear which approach provides the highest discrimination power and how these approaches can be transferred between different datasets and languages. This study aims to compare classification performance based on two types of features: features automatically extracted by a pretrained convolutional neural network (CNN) and HF designed by human experts. Both approaches are evaluated on a multilingual dataset collected from 143 PD patients and 151 healthy controls in the Czech Republic, United States, Colombia, and Hungary. The subjects performed the spiral drawing task (SDT; a language-independent task) and the sentence writing task (SWT; a language-dependent task). Models based on logistic regression and gradient boosting were trained in several scenarios, specifically single language (SL), leave one language out (LOLO), and all languages combined (ALC). We found that the HF slightly outperformed the CNN-extracted features in all considered evaluation scenarios for the SWT. In detail, the following balanced accuracy (BACC) scores were achieved: SL-0.65 (HF), 0.58 (CNN); LOLO-0.65 (HF), 0.57 (CNN); and ALC-0.69 (HF), 0.66 (CNN). However, in the case of the SDT, features extracted by a CNN provided competitive results: SL-0.66 (HF), 0.62 (CNN); LOLO-0.56 (HF), 0.54 (CNN); and ALC-0.60 (HF), 0.60 (CNN). In summary, regarding the SWT, the HF outperformed the CNN-extracted features over 6% (mean BACC of 0.66 for HF, and 0.60 for CNN). In the case of the SDT, both feature sets provided almost identical classification performance (mean BACC of 0.60 for HF, and 0.58 for CNN).