I have a bit of experience of this, my dad had hep B.
Most people who are carriers don't exhibit symptoms, however some people get chronic hepatitis which usually leads to death, through liver failure or some related illness.
The blood test is to check the vaccine has worked, they will be looking for antibodies to hep B to prove the vaccine has worked, rather than the actual virus.
I don't think you're at a good risk of being infected through your parents, although hep B is 100 times more infectious than HIV, i'm sure they must have periodic tests to check they're clean as they're nurses and in contact with the sick?