To reduce the processing delay from the sequentially running virtual network functions (VNFs) in a service function chain (SFC), network function parallelism (NFP) is introduced that allows VNFs of the SFC to run in parallel. Existing NFP solutions only focused on improving parallelism benefits without paying much attention to resource utilization while deploying VNFs of SFCs. We take advantage of resource-delay dependency to propose a flexible and efficient parallelized SFC placement mechanism called FlexSFC which determines the optimal SFC placement while reducing resource usage and meeting end-to-end delay guarantees of the SFCs deployed. Initial results show that FlexSFC guarantees the end-to-end delay requirement with better resource utilization and SFC acceptance rate than the state-of-the-art approaches. © 2022 IEEE.