This problem is the Critical Path Method, a technique used in project management to find the minimum time to complete a project with dependencies. The longest path in the dependency graph is the critical path. It determines the project duration. You cannot finish faster because this path has no slack time. Any delay on the critical path delays the entire project.
You've now seen three DP patterns on DAGs: longest path (Longest Flight Route), counting paths (Game Routes), and critical path (Parallel Courses III). The template is always the same: topological order plus DP transitions. The only difference is the operation (max, sum, or weighted max).