While the quality of parts produced by additive manufacturing is generally evaluated by surface roughness, relative density, and mechanical properties, the issue of dimensional accuracy is not examined sufficiently. However, dimensional accuracy is very important for the final use and finishing of a product. Since the dimensional change mainly occurs due to shrinkage resulting from the heat energy applied during the sintering process, the effect of production parameters in the additive manufacturing method is quite large. To minimize shrinkage and increase dimensional accuracy, manufacturing parameters need to be optimized and meticulously examined. This study was aimed at determining the effects of manufacturing parameters on geometric tolerances in the production of parts using the additive manufacturing method. AlSi10Mg powder alloy and selective laser melting (SLM) technology were used in the additive manufacturing of this alloy in part production. Twelve different laser powers and scanning speeds, as well as fixed scanning range and layer thickness parameters, were used in production. In determining geometric tolerances, features such as hole diameter change, deviation from angularity, deviation from perpendicularity, deviation from flatness, and deviation from parallelism were taken into consideration. As a result of the study, deviation values increased in high and low laser power/scanning speed combinations. Minimum deviation amounts were obtained in the range of 250–310 laser power and 785–974 scanning speed, which are the middle values of the parameters used. The optimum values of different output responses have been obtained with different production parameters, but for the final use and quality control approval of the product, it is necessary to determine the input parameters at which all output responses are optimal. In this process, the gray relational analysis optimization method, which is one of the multi-criteria decision-making methods, was preferred. As a result of the optimization, the optimum manufacturing parameters for geometric tolerances were determined as the 290/911 laser power/scanning speed combination.