Summary: Many combinatorial problems can be efficiently solved for partial \(k\)-trees (graphs of treewidth bounded by \(k\)). The edge-coloring problem is one of the well-known combinatorial problems for which no efficient algorithms were previously known, except a polynomial-time algorithm of very high complexity. This paper gives a linear-time sequential algorithm and an optimal parallel algorithm which find an edge-coloring of a given partial \(k\)-tree with the minimum number of colors for fixed \(k\).