Parkinson’s disease is a neurodegenerative disorder characterized by a variety of motor and non-motor symptoms. Particularly, several speech impairments appear in the initial stages of the disease, which affect aspects related to respiration and the movement of muscles and limbs in the vocal tract. Most of the studies in the literature aim to assess only one specific task from the patients, such as the classification of patients vs. healthy speakers, or the assessment of the neurological state of the patients. This study proposes a multitask learning approach based on convolutional neural networks to assess at the same time several speech deficits of the patients. A total of eleven speech aspects are considered, including difficulties of the patients to move articulators such as lips, palate, tongue and larynx. According to the results, the proposed approach improves the generalization of the convolutional network, producing more representative feature maps to assess the different speech symptoms of the patients. The multitask learning scheme improves in of up to 4% the average accuracy relative to single networks trained to assess each individual speech aspect.