Do parents have a responsibility to undergo prenatal tests?

You decide. Prenatal testing varies in its scope. Some simply verifies the health of the pregnant lady or the chance need for some nutritional support. When looking for fetal problems, many parents would want to know if they should deliver in a facility with advanced services to handle a problem kid. Some would want to prepare emotionally for a child with defects or have the opportunity to end the pregnancy.
Good question. I believe so, however they are not mandatory. It depends on moral, cultural, spiritual, religious and socioeconomic facotrs, and hinges on what they will do with the information gleaned from such tests. Genetic counseling can help parents decide what is best for their family.