This study investigates data transmission in a virtualized radio access network (RAN) architecture, where a Central Unit (CU) is interconnected with Remote Units (RUs) via a mid-haul Passive Optical Network (PON), and user equipments (UEs) are wirelessly served by the RUs over the air interface. In this architecture, data transmission follows a two-stage process. Stage 1: The Central Unit (CU) buffers incoming data until the scheduler triggers transmission to the Remote Units (RUs) via the mid-haul Passive Optical Network (PON), which is subject to capacity constraints. Stage 2: Each RU then delivers the data to user equipments (UEs) over the air interface, with wireless transmission rates that exhibit time-varying characteristics. The objective is to maximize the aggregate utility of all users, where utility is modeled as a function of long-term data rates. This problem was first investigated by (Sinha et al. 2019), who proposed a linear programming (LP)-based 1/2-approximation algorithm. Subsequently, (Sinha et al. 2019) demonstrated that the problem can be reduced to a submodular maximization problem with partition matroid constraints, and accordingly developed a greedy-based 1/2-approximation algorithm. In this study, we focus on parallel algorithms design for this problem and propose an improved parallel algorithm with an approximation ratio approaching \(1/(1+\alpha )\) , where \(\alpha \) denotes the curvature parameter of the submodular utility function.