See that P$^{NP}_{||} = P^{NP}_{O(\log n)}$
Please refer the theorem 7, in On Truth-Table Reducibility to SAT
and the Difference Hierarchy over NP  and On Truth-Table Reducibility to SAT by
Samuel R. Buss. The problem you mentioned is the original result by Buss. Hemachandra proved one of the inclusions in his PhD thesis.