Recently proposed automatic pathological speech classification techniques use unsupervised auto-encoders to obtain a high-level abstract representation of speech. Since these representations are learned based on reconstructing the input, there is no guarantee that they are robust to pathology-unrelated cues such as speaker identity information. Further, these representations are not necessarily discriminative for pathology detection. In this paper, we exploit supervised auto-encoders to extract robust and discriminative speech representations for Parkinson's disease classification. To reduce the influence of speaker variabilities unrelated to pathology, we propose to obtain speaker identity-invariant representations by adversarial training of an auto-encoder and a speaker identification task. To obtain a discriminative representation, we propose to jointly train an auto-encoder and a pathological speech classifier. Experimental results on a Spanish database show that the proposed supervised representation learning methods yield more robust and discriminative representations for automatically classifying Parkinson's disease speech, outperforming the baseline unsupervised representation learning system.