Almost one half of celiac patients have no symptoms at all, but they have intestinal changes, antibodies in the blood, and anemia. You can be really sure, your diagnosis is accurate, especially blood test is reliable one.
Your parents *may* have celiac disease without any symptoms, but if they were not tested, this can't be said. It is not the disease which is inherited, but predisposition for the disease is. A person is predisposed for celiac disease if he/she have one specific DNA change, which can be tested. Having this change does not necessary result in disease. If one member of a family has celiac disease, about 1 of ten other familly members is likely to have it.
Please don't follow a regular diet. If you do, you may end up with long term problems that you wouldn't wish on your worst enemy. People with undiagnosed celiac can have problems that can end up looking like MS or other disorders and the symptoms you'd end up with down the line would be horrible.
Please, please, please start on a GF diet today and stick to it. There is no such thing as mild celiac - it's all-or-nothing. I only wish my husband had found out years ago about his condition. His condition didn't turn up in the blood tests, and if those are how your parents were checked they should be rechecked via fecal antibody test. It's a much better way of detecting those who have few symptoms.