Slices were introduced in 5G to enable the co-existence of applications with different requirements on a single infrastructure. Slices may be delay-constrained for mission-critical applications such as Tactile Internet applications. When delay-constrained slices are implemented as collections of virtual network function (VNF) chains, a key challenge is to place the VNFs and route the traffic through the chains to meet a strict delay constraint. Parallel VNF processing has been proposed as a promising approach. However, this approach increases the number of physical nodes in the chains, and thus decreases the reliability, which is also critical for Tactile Internet applications. Furthermore, the cost depends upon the specific VNF placement and traffic routing, as nodes and links are heterogeneous. This paper tackles the issues of reliability and cost when embedding delay-constrained slices. We model the problem as an optimization problem that minimizes reliability degradation and cost while ensuring the strict delay constraint when a parallel VNF processing approach is used. Due to the complexity of the formulated problem, we also propose a Tabu search-based algorithm to find sub-optimal solutions. The results indicate that our proposed algorithm can significantly improve cost and reliability while meeting a strict delay constraint.