For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: $ODD^A_n = \{(x_1, \dots ,x_n): {\tt\#}^A_n(x_1, \dots , x_n) \text{is odd}\}$ and, for $m \geq 2,$ $\text{MOD}m^A_n = \{(x_1, \dots ,x_n):{\tt\#}^A_n(x_1, \dots ,x_n) \not\equiv 0 (\text{mod} m)\},$ where ${\tt\#}^A_n(x_1, \dots ,x_n) = A(x_1)+\cdots+A(x_n)$. (We identify A(x) with $\chi_A (x)$, where $\chi_A$ is the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide $\text{ODD}^A_n$ and $\text{MOD}m^A_n: \bullet\text{ODD}^A_n$ can be decided with n parallel queries to A, but not with n - 1. $\bullet \text{ODD}^A_n$ can be decided with $\lceil log(n + 1)\rceil$ sequential queries to A but not with $\lceil log(n + 1)\rceil - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil n/m\rceil + \lfloor n/m\rfloor$ parallel queries to A but not with $\lceil n/m\rceil + \lfloor n/m\rfloor - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil$ sequential queries to A but not with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil - 1$. The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that $\text{ODD}^A_n$ cannot be decided with n - 1 parallel queries to A. Since every truth-table degree also contains a set B such that $\text{ODD}^B_n$ can be decided with one query to B, a set's query complexity depends more on its structure than on its degree. For a fixed set A, $Q(n,A) = \{S : S \text{can be decided with n sequential queries to} A\},\\Q_\parallel(n, A) = \{S : S \text{can be decided with n parallel queries to} A\}.$ We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: $\bullet Q(0,A) \subset Q(1,A) \subset Q(2,A) \subset\cdots\\ \bullet Q_\parallel(0, A) \subset Q_\parallel(1, A) \subset Q_\parallel(2,A) \subset\cdots$ The same is true if A is a jump.