Are there any examples of complex statistics that are too complex to be computed directly and need simulations (Parametric Bootstrap from a small dataset) rather than computing the statistics from the estimated parametric model?
Here's an example of something that's hard to do "directly" but seems reasonably amenable to simulation:
A prediction interval for a sum of forecasts from a Poisson GLM with several predictors.
Difficulties can be seen in several ways, but here's one: a common way to construct prediction intervals is by constructing a pivotal quantity, but there aren't any - only approximate (asymptotic) pivots, but many problems involve small counts.
(The lack of a pivotal quantity also makes it hard to construct exchangeable quantities suitable for the more usual kinds of bootstrap.)
[There are a number of other issues with this particular problem, but I won't labour the point.]
This maybe a bit arcane, but bootstrap is used in shape analysis for hypothesis testing. Null hypothesis is that the average shapes for two groups are the same. For example, in works by Vic Patrangenaru, see one of his papers here http://arxiv.org/pdf/math/0507423.pdf