In a previous result, the authors showed that a clique tree of a chordal graph can be constructed in O(log n) parallel computing time with O(n3) processors on CRCW PRAM, where n is the number of nodes of the graph. In this paper, it will be shown that this result can be extended in two ways. First, we show that from the parallelcliquetree constructing algorithm, we can derive an exact formula of countingcliquetrees of a labeled connected chordal graph. Next, we show that a perfecteliminationscheme of a chordal graph can be computed in O(log n) time with O(n2) processors on CREW PRAM once a cliquetree of the graph is given. This implies that a perfecteliminationscheme of a chordal graph can be computed in O(log n) time with O(n3) processors on CRCW PRAM.