Multi-criteria scheduling problems under uncertainty remain a relatively unexplored topic in theoretical computer science despite substantial practical interests. This work studies a bi-objective identical parallel machine scheduling problem under uncertainty, in which the first objective is to minimize the total completion time, and the second is to minimize the makespan. Especially a job's processing time is assumed to be represented by a polynomial function with respect to scenario u∈U, where U⊂R+ is an interval containing an infinite number of scenarios. In this work, we are looking for a compact and complete description of the set of possibly optimal solutions, along with their objective function values, over the set of scenarios or an approximation with a performance guarantee. First, to better understand the characteristics of the studied problem, we consider two single-objective problems: parallel machine scheduling problem under uncertainty with the total completion time and makespan criterion, respectively. For the problem with the total completion time criterion, we demonstrate that the set of possibly optimal schedules can be found in polynomial time. In contrast, for the problem with the makespan criterion, we provide a (1+ϵ)-approximation algorithm. For the bi-objective problem, we provide a 2-approximation algorithm for any number of parallel machines and a (1+ϵ)-approximation algorithm where a fixed number of parallel machines is considered.