We consider a project that consists of activities operating in parallel under various temporal constraints, including start-to-start, start-to-finish and finish-to-start precedence relations, early-start, late-start and late-finish time boundaries, and due dates. Scheduling problems are formulated to find optimal schedules for the project with respect to different objective functions to be minimized, including the project makespan, the maximum deviation from the due dates, the maximum flow-time, and the maximum deviation of finish times. We represent the problems as optimization problems in terms of tropical mathematics, and then solve these problems by applying direct solution methods of tropical optimization. As a result, new direct solutions of the problems are obtained in a compact vector form, which is ready for further analysis and practical implementation.