painting fence during exceptionally rainy summer
A better approach would be to set up some sort of cover, (if feasible size wise). Make sure there is air flow, i.e. don't just drape a tarp or polythene over the fence. I've got loads of quick release clamps and often use them on top of things to create a wider support for a tarp to hang clear of the surface. 

It may take a week or two for the fence to dry out if it's soaked through.

Otherwise I agree with Chris, it needs to be dry. Sure, a water based preservative might tolerate more moisture in the wood than say an oil based one but still, dry is best. If you have access to a moisture meter, give it a try. Between 10% and 15% is ideal (assuming normal weather/humidity), and never paint anything over around 20%.