A new parallel algorithm for the prefix minima problem is presented for inputs drawn from the range of integers $[1..s]$. For an input of size $n$, it runs in $O(\log\log\log s)$ time and $O(n)$ work (which is optimal). A faster algorithm is presented for the special case $s=n$; it runs in $O(\log ^*n)$ time with optimal work. Both algorithms are for the Priority concurrent-read concurrent-write parallel random access machine (CRCW PRAM). A possibly surprising outcome of this work is that, whenever the range of the input is restricted, the prefix minima problem can be solved significantly faster than the $\Omega( \log\log n )$ time lower bound in a decision model of parallel computation, as described by Valiant [SIAM J. Comput., 4 (1975), pp. 348--355]. The top-bottom routing problem, which is an important subproblem of routing wires around a rectangle in two layers, is also considered. It is established that, for parallel (and hence for serial) computation, the problem of top-bottom routing is no harder than the prefix minima problem with $s=n$, thus giving an $O(\log ^*n)$ time optimal parallel algorithm for the top-bottom routing problem. This is one of the first nontrivial problems to be given an optimal parallel algorithm that runs in sublogarithmic time.