By "in some sense", do you mean it can be parallelized to O(1) time as long as you don't have to actually build up the data structures that result from partitioning (e.g. arrays)? Then it's not clear to me why this assumption is applicable to quicksorting an array. Doesn't it call for some non-obvious representation of intermediate results?
But isn't recursion depth the same as depth if each operation in your recursive function is constant time (except for the recursing bits)? Thus you can look at recursive depth as computing depth modulo the bits you compute in your function at a fixed level. I was speculating that partitioning would require O(log n) parallel steps. Can you give details or provide a link to achieving it constant depth?
Quicksort partitioning normally takes O(N) time, but it can be in some sense perfectly parallelized once you have the pivot.