Pre-trained language models (PrLMs) have demonstrated superior performance due to their strong ability to learn universal language representations from self-supervised pre-training. However, even with the help of the powerful PrLMs, it is still challenging to effectively capture task-related knowledge from dialogue texts which are enriched by correlations among speaker-aware utterances. In this work, we present SPIDER, Structural Pre-traIned DialoguE Reader, to capture dialogue exclusive features. To simulate the dialogue-like features, we propose two training objectives in addition to the original LM objectives: 1) utterance order restoration, which predicts the order of the permuted utterances in dialogue context; 2) sentence backbone regularization, which regularizes the model to improve the factual correctness of summarized subject-verb-object triplets. Experimental results on widely used dialogue benchmarks verify the effectiveness of the newly introduced self-supervised tasks.