Despite the various attractive features that grid computing has to offer, it has many great security challenges, such as access control. With the expansion of the network scale, a large number of authorization requests have to be treated; on the other hand, the multi-domain nature of grid computing generates difficult to manage questions about cross-domain access control, and a variety of solutions use the role mapping mechanism to allow collaborations between domains. But this mechanism gives a potential risk of violating consistency properties of domains. This article aims to address this issue and proposes a parallel access control model in cross-domain grid computing architecture to be more convenient to the security requirements of the multi-domain environment. Finally, as a proof of concept, the authors implement a cross-domain and parallel authorization simulator (CD-PAS) where experiments are done. The obtained results show that the proposed model is sensitive to the number of authorization requests. In addition, it can effectively reduce the execution time of access control mechanism.