In multi-access edge computing (MEC) networks, parallelized service function chains (P-SFCs) can provide low-delay network services for mobile users by deploying virtualized network functions (VNFs) to process user requests in parallel. These VNFs are unreliable due to software faults and server failures. A practical way to address this is to deploy idle backup VNFs (BVNFs) near these active VNFs and activate them when active VNFs fail. However, deploying BVNFs preempts server resources and decreases the number of accepted user requests. Thus, this paper proposes a reliability enhancement approach that uses BVNFs satisfying the delay requirement as active VNFs to form P-SFCs, which contributes to the delay reduction and reliability enhancement. Since the resource capacities of edge servers can only deploy a certain number of P-SFCs and BVNFs, establishing how to deploy the minimum number of P-SFCs and BVNFs to satisfy the delay and reliability requirements of mobile users and maximize the number of accepted user requests is a challenging problem. In this paper, we first model the dynamics of delay and reliability caused by VNF parallelization and BVNFs deployment, then formulate the joint deployment problem of P-SFCs and BVNFs. Next, we design an approximation algorithm to deploy critical VNFs and BVNFs on a target edge server and schedule the data traffic of user requests processed by P-SFCs. Experimental results based on real-world datasets show that our proposed algorithm outperforms two benchmark algorithms in terms of throughput, delay, reliability, and resource utilization.