Network function virtualization (NFV) is a promising solution to realize a variety of network services. By definition, virtual network functions (VNFs) are chained together to realize different services. However, chaining is not an ideal solution as service latency grows linearly with respect to the length of the chain. Motivated by the fact that many VNFs can be parallelized, we investigate parallelism of VNFs for acceleration. The dependency of the VNFs is characterized by a directed acyclic graph (DAG). We aim to deploy the VNFs in the right place and process them in parallel without violating the DAG, to minimize the overall delay. However, directly solving the delay minimization problem is NP-hard, and it may also introduce a large number of duplicated packets to burden the system. To deal with these issues, we propose the Prune and Plant (P&P) scheme with polynomial computational complexity, to reduce the overall delay while limiting the number of duplicated packets. P&P comprises two stages: in the Prune stage, we prune the original DAG into a series-parallel graph (SP-graph), which eliminates NP-hardness while maintaining parallelism of VNFs. In the Plant stage, we find the optimal placement for the VNFs with respect to the SP-graph. By both simulation and prototyping, we demonstrate that P&P significantly outperforms benchmark schemes.