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Nonlinear integer programs involve optimizing nonlinear objectives with variables restricted to integer values, and have widespread applications in areas such as resource allocation and portfolio selection. One approach to solving these problems is the augmentation procedure, which iteratively refines a feasible solution by identifying augmenting steps from the Graver Basis--a set of test directions. While this method guarantees termination in polynomially many steps, computing the Graver Basis exactly is known to be \(\mathcal{NP}\)-hard. To address this computational challenge, we propose Multi-start Augmentation via Parallel Extraction (MAPLE), a GPU-based heuristic designed to efficiently approximate the Graver Basis. MAPLE extracts test directions by optimizing non-convex continuous problems, leveraging first-order methods to enable parallelizable implementation. The resulting set of directions is then used in multiple augmentations, each seeking to improve the solution's optimality. The proposed approach has three notable characteristics: (i) independence from general-purpose solvers, while ensuring guaranteed feasibility of solutions; (ii) high computational efficiency, achieved through GPU-based parallelization; (iii) flexibility in handling instances with shared constraint matrices but varying objectives and right-hand sides. Empirical evaluations on QPLIB benchmark instances demonstrate that MAPLE delivers performance comparable to state-of-the-art solvers in terms of solution quality, while achieving significant gains in computational efficiency. These results highlight MAPLE's potential as an effective heuristic for solving nonlinear integer programs in practical applications.