A new method is developed to schedule jobs on parallel machines with availability constraints. The objective of the problem is to minimize the makespan of the total production schedule. Without the availability constraints the scheduling of machines is a Pm || Cmax problem. The scheduling of this problem was the topic of many earlier papers. The main contribution of this research is that the schedule of the jobs on parallel machines with availability constraints is determined within a single implicit enumer- ation algorithm. Within the general enumeration scheme, the loads of each machine are enumerated in a lexicographic order. An exact integer linear programming model is provided, too. The difficulty of the problem depends on the properties of the pro- cessing times, the number of machines, and the number of availability constraints on the machines. In some subclasses, problems with very large number of jobs are solved. The largest problems solved within one hour limit have 1, 000, 000 jobs.