Past research has shown that masonry mesoscale descriptions, where bricks and mortar joints are modelled separately, offer a realistic representation of the mechanical behaviour of masonry components. In the case of masonry arches, thus far the use of this approach has been restricted to 2D analysis mainly because of the significant computational effort required. However conventional 2D models may lead to a crude representation of the response of masonry arches which is inherently three-dimensional, and they cannot properly capture the actual response of masonry arches subjected to eccentric loading or the behaviour of arches with complex geometry (e.g. skew arches). In this paper, the nonlinear response of brick-masonry arches up to collapse is investigated using an accurate 3D mesoscale description utilising solid elements for representing brick units and 2D nonlinear interface elements for describing mortar joints and brick-mortar interfaces. The masonry mesoscale strategy is then combined with a domain partitioning approach allowing for parallel computation which guarantees computational efficiency. The accuracy and potential of the proposed numerical description are shown in numerical examples, including comparisons against experimental results on realistic square and skew brick-masonry arches.