As an effective technique to relieve the problem of resource constraints on mobile devices (MDs), the computation offloading utilizes powerful cloud and edge resources to process the computation-intensive tasks of mobile applications uploaded from MDs. In cloud-edge computing, the resources (e.g., cloud and edge servers) that can be accessed by mobile applications may change dynamically. Meanwhile, the parallel tasks in mobile applications may lead to the huge solution space of offloading decisions. Therefore, it is challenging to determine proper offloading plans in response to such high dynamics and complexity in cloud-edge environments. The existing studies often preset the priority of parallel tasks to simplify the solution space of offloading decisions, and thus the proper offloading plans cannot be found in many cases. To address this challenge, we propose a novel real-time and Dependency-aware task Offloading method with Deep Q-networks (DODQ) in cloud-edge computing. In DODQ, mobile applications are first modeled as Directed Acyclic Graphs (DAGs). Next, the Deep Q-Networks (DQN) is customized to train the decision-making model of task offloading, aiming to quickly complete the decision-making process and generate new offloading plans when the environments change, which considers the parallelism of tasks without presetting the task priority when scheduling tasks. Simulation results show that the DODQ can well adapt to different environments and efficiently make offloading decisions. Moreover, the DODQ outperforms the state-of-art methods and quickly reaches the optimal/near-optimal performance.