We consider a stochastic parallel machine scheduling problem, where the jobs have uncertain processing time described by a normal probability distribution. The objective is to maximize the probability that all the jobs are completed before a common due date. The considered problem has many practical applications, but it is notoriously known to be difficult as it involves several non-linearities which complicates its analysis and solution.In this work, we have developed novel lower and upper bounds on the objective function. The upper bound is computed via a solution to a problem where a subset of machines is represented as a single machine having a modified due date. Furthermore, we study lower and upper bounds on the number of jobs that must be scheduled on a machine in an optimal schedule. Subsequently, we use the bounds to construct an efficient branch-and-price algorithm where the pricing problem is found to be related to an inflatable stochastic Knapsack problem. An advantage of the branch-and-price algorithm is a constraint branching mechanism that mitigates symmetries in the solution space. The performance evaluation of the proposed algorithm shows that our algorithm outperforms the state-of-the-art method.In this paper, we also study a special case of the problem assuming two machines. We developed a scalable method whose efficiency arises from the concavity of the relaxed objective function and a fast procedure to recover optimal integer solution from it. These improvements allowed us to solve instances with 500 jobs within a few seconds.