Women don't have periods once they become pregnant, so no the baby is not "drinking the blood".
To become pregnant, an egg needs to be released (ovulation) at some point in a cycle. If the egg is not fertilized (ie - NOT pregnant) the body sheds the egg as well as the uterine lining that had prepared for a fertilized egg during the cycle. That is your period. If the egg IS fertilized (ie - pregnant), then the egg implants into the uterine lining and obviously is NOT shed via period - instead, it is the beginning of your pregnancy. As long as you are pregnant, you will not get a period since you will no longer be ovulating. The production of the hormone Progesterone is important during a cycle for preparing the uterine lining for a pregnancy, and it also prevents the release of any other eggs throughout pregnancy. As long as you are pregnant, you will produce enough Progesterone to do it's job and prevent ovulation.