With the development of the Internet of Things (IoT), certain IoT devices have the capability to not only accomplish their own tasks but also simultaneously assist other resource-constrained devices. Therefore, this paper considers a device-assisted mobile edge computing system that leverages auxiliary IoT devices to alleviate the computational burden on the edge computing server and enhance the overall system performance. In this study, computationally intensive tasks are decomposed into multiple partitions, and each task partition can be processed in parallel on an IoT device or the edge server. The objective of this research is to develop an efficient online algorithm that addresses the joint optimization of task partitioning and parallel scheduling under time-varying system states, posing challenges to conventional numerical optimization methods. To address these challenges, a framework called online task partitioning action and parallel scheduling policy generation (OTPPS) is proposed, which is based on deep reinforcement learning (DRL). Specifically, the framework leverages a deep neural network (DNN) to learn the optimal partitioning action for each task by mapping input states. Furthermore, it is demonstrated that the remaining parallel scheduling problem exhibits NP-hard complexity when considering a specific task partitioning action. To address this subproblem, a fair and delay-minimized task scheduling (FDMTS) algorithm is designed. Extensive evaluation results demonstrate that OTPPS achieves near-optimal average delay performance and consistently high fairness levels in various environmental states compared to other baseline schemes.