We consider the parallel variable distribution (PVD) approach proposed by Ferris and Mangasarian [SIAM J. Optim., 4 (1994), pp. 815--832] for solving optimization problems. The problem variables are distributed among p processors with each processor having the primary responsibility for updating its block of variables while allowing the remaining "secondary" variables to change in a restricted fashion along some easily computable directions. For constrained nonlinear programs, convergence in [M. C. Ferris and O. L. Mangasarian, SIAM J. Optim., 4 (1994), pp. 815--832] was established in the special case of convex block-separable constraints. For general (inseparable) constraints, it was suggested that a dual differentiable exact penalty function reformulation of the problem be used. We propose to apply the PVD approach to problems with general convex constraints directly and show that the algorithm converges, provided certain conditions are imposed on the change of secondary variables. These conditions are both natural and practically implementable. We also show that the original requirement of exact global solution of the parallel subproblems can be replaced by a less stringent sufficient descent condition. The first rate of convergence result for the class of constrained PVD algorithms is also given.