See that P$^{NP}_{||} = P^{NP}_{O(\log n)}$ I'm trying to prove that P$^{NP}_{||} =$ P$^{NP}_{O(\log n)}$ where $n$ is the length of the input.

So, to see that polynomially many non-adaptive queries to a problem in NP can do as much as logarithmically many adaptive ones.

I know I need to see both inclusions, and have worked out the "easy" one: P$^{NP}_{||} \supseteq$ P$^{NP}_{O(\log n)}$ by arranging all the logarithmically many possibilities in a long deterministic one, and then use the fact that it's still polynomial (from known time complexity properties).

Now, I bang my head on a wall when I try to do the other one. I'm not even sure of where to start. I would very much appreciate your input on this problem, and besides the problem advice on where I could read examples on similar problems to get a better hang on them.
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.