Could diabetes be analogous to Parkinson's? Diabetics 
    generally have lost the ability to process carbs (usually 
    by eating too many);

No.

I'm assuming you are talking about type 2, which has nothing to do with the ability to process carbs but instead resistance to insulin. This is caused by so many different factors like genes, medications, stress, smoking, high intake of certain types of fats, lack of exercise, white rice, pollution, etc.

You can't be talking about type 1 because you'd be absurdly wrong if you were.

Source: parent of type 1 diabetic