The drink parallel of famished is parched.
Dictionary.com main section
5. verb (used without object) to suffer from heat, thirst, or need of water.
Dictionary.com "World English Dictionary" section
2. (tr; usually passive) to make very thirsty: I was parched after the run
Dehydrated is a more clinical term, and would be parallel to malnourished, I think.
Both famished and parched can be used in both formal and informal contexts, although they're frequently used with a bit of hyperbole implied, which skews them toward the informal.
I'm so parched I think I could drink a river.
There are also a lot of idioms out there to signify extreme thirst, such as the "spitting cotton" one mentioned by CopperKettle, but if you're looking for the most straightforward parallel to famished, then you really can't do better than parched.