The partial Latin square extension problem is to fill as many as possible empty cells of a partially filled Latin square. This problem is a useful model for a wide range of applications in diverse domains. This paper presents the first massively parallel evolutionary algorithm algorithm for this computationally challenging problem based on a transformation of the problem to partial graph coloring. The algorithm features the following original elements. Based on a very large population (with more than $10^4$ individuals) and modern graphical processing units, the algorithm performs many local searches in parallel to ensure an intensive exploitation of the search space. The algorithm employs a dedicated crossover with a specific parent matching strategy to create a large number of diversified and information-preserving offspring at each generation. Extensive experiments on 1800 benchmark instances show a high competitiveness of the algorithm compared to the current best performing methods. Competitive results are also reported on the related Latin square completion problem. Analyses are performed to shed lights on the roles of the main algorithmic components. The code of the algorithm will be made publicly available.