Hi Amelie,
I home-cook for my dog using Dr. Strombeck's book, too. Carbohydrates play an important role in a home-cooked diet by providing an energy source for the body so the protein source can be exclusively used for tissue repair and growth. Carbohydrates also provide nutrients and varying amounts and types of fiber depending on the source.
Brown rice and oatmeal are more calorie dense and higher in fat and protein than white rice and pasta. They both contain higher amounts of most nutrients. Brown rice is higher in fiber. Both are much higher in phosphorous than white rice with similar amounts of calcium.
If you want to compare their nutrient content, you can do a food search in the USDA Nutrient Database:
http://www.nal.usda.gov/fnic/foodcomp/search/
I lean towards oatmeal, nutrientwise, as long as the dog agrees with me.
