We present parallel approximation algorithms for maximization problems expressible by integer linear programs of a restricted syntactic form introduced recently by Barland et al. One of our motivations was to show whether the approximation results in the framework of Barland et al. holds in the parallel seeting. Our results are a confirmation of this, and thus we have a new common framework for both computational settings. Also, we prove almost tight non-approximability results, thus solving a main open question of Barland et al. We obtain the results through the constraint satisfaction problem over multi-valued domains (which is a natural generalization of boolean constraint satisfaction and has additional relations to other problems), for which we show non-approximability results and develop parallel approximation algorithms. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of Probastically Checkable Proofs and Multi-Prover One-Round Proof Systems.