Automatic assessment of dysarthric speech is essential for sustained treatments and rehabilitation. However, obtaining atypical speech is challenging, often leading to data scarcity issues. To tackle the problem, we propose a novel automatic severity assessment method for dysarthric speech, using the self-supervised model in conjunction with multi-task learning. Wav2vec 2.0 XLS-R is jointly trained for two different tasks: severity classification and auxiliary automatic speech recognition (ASR). For the baseline experiments, we employ hand-crafted acoustic features and machine learning classifiers such as SVM, MLP, and XGBoost. Explored on the Korean dysarthric speech QoLT database, our model outperforms the traditional baseline methods, with a relative percentage increase of 1.25% for F1-score. In addition, the proposed model surpasses the model trained without ASR head, achieving 10.61% relative percentage improvements. Furthermore, we present how multi-task learning affects the severity classification performance by analyzing the latent representations and regularization effect.