Secret key agreement from correlated physical layer observations is a cornerstone of information-theoretic security. This paper proposes and rigorously analyzes a complete, constructive protocol for secret key agreement from Gaussian sources using Sparse Regression Codes (SPARCs). Our protocol systematically leverages the known optimality of SPARCs for both rate-distortion and Wyner-Ziv (WZ) coding, facilitated by their inherent nested structure. The primary contribution of this work is a comprehensive end-to-end analysis demonstrating that the proposed scheme achieves near-optimal secret key rates with strong secrecy guarantees, as quantified by a vanishing variational distance. We explicitly characterize the gap to the optimal rate, revealing a fundamental trade-off between the key rate and the required public communication overhead, which is governed by a tunable quantization parameter. Furthermore, we uncover a non-trivial constrained optimization for this parameter, showing that practical constraints on the SPARC code parameters induce a peak in the achievable secret key rate. This work establishes SPARCs as a viable and theoretically sound framework for secure key generation, providing a compelling low-complexity alternative to existing schemes and offering new insights into the practical design of such protocols.