We consider the problem of finding the optimal parallel schedule (also called free schedule) for a Uniform Recurrence Equation(URE) over an arbitrary family of integral polyhedra. Many computation structures such as nested loops with constant dependence vectors and systolic algorithms can be described by UREs. An optimal schedule is a function which schedules computations as soon as possible. We show that for a URE over a family of integral polyhedra, the free schedule is bounded within a constant to a piecewise quasi-linear function for computations not too "close" to the boundary, provided that the URE has a quasi-linear schedule. Furthermore, the piecewise quasi-linear function itself is a valid schedule and can be obtained by a parametric rational linear programming. We also show that it is always possible to choose a single quasi-linear schedule which is "almost" optimal with respect to the minimal execution time of the last computation over any domain in a family of bounded integral polyhedra. Intuitively, the result shows that we can effectively find a quasi-linear schedule which is "almost" optimal. This justifies using (quasi)-linear schedules in the parallelization of nested loops with constant dependence vectors and in the systolic array synthesis. The result is an extension to a result given in {KMW67].