Conversational semantic role labeling (CSRL) is believed to be a crucial step towards dialogue understanding. However, it remains a major challenge for existing CSRL parser to handle conversational structural information. In this paper, we present a simple and effective architecture for CSRL which aims to address this problem. Our model is based on a conversational structure aware graph network which explicitly encodes the speaker dependent information. We also propose a multi-task learning method to further improve the model. Experimental results on benchmark datasets show that our model with our proposed training objectives significantly outperforms previous baselines.