Summary: We consider parallel identical machines scheduling problems with deteriorating maintenance activities and the option of job rejection. Each machine has at most one deteriorating maintenance activity. The length of each maintenance activity increases linearly with its starting time. The location of the maintenance activity on each machine needs to be determined. The goal is to find the sequence of jobs to minimize scheduling cost; we further the model by allowing job rejection. A job is either accepted and processed on one of machines, or rejected. The goal is to determine the sequence of the accepted jobs to minimize scheduling cost of the accepted jobs plus total rejection penalty of the rejected jobs. When the scheduling cost is the makespan, we design a pseudo-polynomial time algorithm, a 2-approximation algorithm and a fully polynomial time approximation scheme. When the scheduling cost is the total completion time, we provide a polynomial time algorithm for the problem. When the scheduling costs are the total weighted completion time under the agreeable ratio assumption and the maximum lateness, we present pseudo-polynomial time algorithms to solve these problems, respectively.