Canonical decomposition for bipartite graphs, which was introduced by Fouquet, Giakoumakis, and Vanherpe (1999), is a decomposition scheme for bipartite graphs associated with modular decomposition. Weak-bisplit graphs are bipartite graphs totally decomposable (i.e., reducible to single vertices) by canonical decomposition. Canonical decomposition comprises series, parallel, and K+S decomposition. This paper studies a decomposition scheme comprising only parallel and K+S decomposition. We show that bipartite graphs totally decomposable by this decomposition are precisely P₆-free chordal bipartite graphs. This characterization indicates that P₆-free chordal bipartite graphs can be recognized in linear time using the recognition algorithm for weak-bisplit graphs presented by Giakoumakis and Vanherpe (2003).