Recent several years have witnessed the surge of asynchronous (async-) parallel computing methods due to the extremely big data involved in many modern applications and also the advancement of multi-core machines and computer clusters. In optimization, most works about async-parallel methods are on unconstrained problems or those with block separable constraints. In this paper, we propose an async-parallel method based on block coordinate update (BCU) for solving convex problems with <i>nonseparable</i> linear constraint. Running on a single node, the method becomes a novel randomized primaldual BCU for multi-block affinely constrained problems. For these problems, GaussSeidel cyclic primaldual BCU is not guaranteed to converge to an optimal solution if no additional assumptions, such as strong convexity, are made. On the contrary, assuming convexity and existence of a primaldual