The park that is friendliest to rain is easily Epcot
Epcot has many large buildings that have a lot of things to do in them.
- Innoventions
- Wonders of Life
- Mission Space (coming soon)
- Test Track (post show)
- Imagination
- The Land
- Living Seas
Plus...
In all of the countries in World Showcase, every shop and restaurant is either connected or is very close to one another. The only pain is that the entrances to the countries aren't especially close to one another, and there are some large gaps between countries (China - Germany, France - UK, UK - Canada).
Magic Kingdom and MGM are not horrible in the rain either. There are a lot of covered areas in both parks (especially in the MK), and both parks have a lot of attractions that continue to run in inclement weather.
The only park that doesn't work so well in the rain is AK, but that has already been covered.