Of course, the study also notes that affluent people, white people, and males are all more likely to think themselves of sound mind, and those demographics skew Republican. However, even controlling for those factors, the Grand Old Party has a more sanguine opinion of its sanity.
The question is: Does this say more about Republicans/Democrats and mental health or Republicans/Democrats and self-reporting?
Just as importantly, did Gallup really ask thousands of people, "Have you lost your mind?"