As an emerging computing model, edge computing greatly expands the collaboration capabilities of the servers. It makes full use of the available resources around the users to quickly complete the task request coming from the terminal devices. Task offloading is a common solution for improving the efficiency of task execution on edge networks. However, the peculiarities of the edge networks, especially the random access of mobile devices, brings unpredictable challenges to the task offloading in a mobile edge network. In this paper, we propose a trajectory prediction model for moving targets in edge networks without users’ historical paths which represents their habitual movement trajectory. We also put forward a mobility-aware parallelizable task offloading strategy based on a trajectory prediction model and parallel mechanisms of tasks. In our experiments, we compared the hit ratio of the prediction model, network bandwidth and task execution efficiency of the edge networks by using the EUA data set. Experimental results showed that our model is much better than random, non-position prediction parallel, non-parallel strategy-based position prediction. Where the task offloading hit rate is closed to the user’s moving speed, when the speed is less 12.96 m/s, the hit rate can reach more than 80%. Meanwhile, we we also find that the bandwidth occupancy is significantly related to the degree of task parallelism and the number of services running on servers in the network. The parallel strategy can boost network bandwidth utilization by more than eight times when compared to a non-parallel policy as the number of parallel activities grows.