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Self-stabilization in distributed systems is a technique to guarantee convergence to a set of legitimate states without external intervention when a transient fault or bad initialization occurs. Recently, there has been a surge of efforts in designing techniques for automated synthesis of self-stabilizing algorithms that are correct by construction. Most of these techniques, however, are not parameterized, meaning that they can only synthesize a solution for a fixed and predetermined number of processes. In this paper, we report a breakthrough in parameterized synthesis of self-stabilizing algorithms in symmetric rings. First, we develop tight cutoffs that guarantee (1) closure in legitimate states, and (2) deadlock-freedom outside the legitimates states. We also develop a sufficient condition for convergence in silent self-stabilizing systems. Since some of our cutoffs grow with the size of local state space of processes, we also present an automated technique that significantly increases the scalability of synthesis in symmetric networks. Our technique is based on SMT-solving and incorporates a loop of synthesis and verification guided by counterexamples. We have fully implemented our technique and successfully synthesized solutions to maximal matching, three coloring, and maximal independent set problems.