As networks grow larger and more intricate, failures become unavoidable. Interconnection networks need continual and reliable functioning, as well as effective routing methods for transmitting data among processors. Ensuring fault tolerance in routing, especially through independent spanning trees (ISTs), is crucial. The 3-ary n-cube network \((Q^{3}_{n})\) boasts numerous properties such as low vertex degree, regularity, and straightforward implementation. In this paper, We have presented an effective parallel constructive approach for ISTs on \(Q^{3}_{n}\) relying on the ascending sequence. Derived from the ISTs constructed before, we obtain a fault-tolerant routing system that utilizes those as its routing table. Following this, we evaluate the effectiveness of the fault-tolerant routing mechanism through simulated data. Simulated data reveals a progressive escalation in transmission success rates as dimensionality increases, approaching near-perfection at almost \(100\%\) . Its findings can provide a valuable benchmark and offer insights into the efficacy of fault-tolerant communication within a broad spectrum of cube-based systems.