One of the main challenges in synthetic biology lies in maximizing the expression levels of a protein by encoding it with multiple copies of the same gene. This task is often conducted under conflicting evaluation criteria, which motivates the formulation of protein encoding as a multi-objective optimization problem. Recent research reported significant results when adapting the artificial bee colony algorithm to address this problem. However, the length of proteins and the number of copies have a noticeable impact in the computational costs required to attain satisfying solutions. This work is aimed at proposing parallel bioinspired designs to tackle protein encoding in multiprocessor systems, considering different thread orchestration schemes to accelerate the optimization process while preserving the quality of results. Comparisons of solution quality with other approaches under three multi-objective quality metrics show that the proposed parallel method reaches significant quality in the encoded proteins. In addition, experimentation on six real-world proteins gives account of the benefits of applying asynchronous shared-memory schemes, attaining efficiencies of 92.11% in the most difficult stages of the algorithm and mean speedups of 33.28x on a 64-core server-grade system.