Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods often require careful parameter tuning or prior knowledge of the sparsity of the signal and/or noise variance. We propose a tuning-free framework that leverages implicit regularization (IR) from overparameterization to overcome this limitation. Our approach reparameterizes the estimation matrix into factors that decouple the shared row-support from individual vector entries and applies gradient descent to a standard least-squares objective. We prove that with a sufficiently small and balanced initialization, the optimization dynamics exhibit a "momentum-like" effect where the true support grows significantly faster. Leveraging a Lyapunov-based analysis of the gradient flow, we further establish formal guarantees that the solution trajectory converges towards an idealized row-sparse solution. Empirical results demonstrate that our tuning-free approach achieves performance comparable to optimally tuned established methods. Furthermore, our framework significantly outperforms these baselines in scenarios where accurate priors are unavailable to the baselines.