A person gets their blood from both parents. You are going to have to explain it in terms easy for your parents as this will certainly involve overcoming some deep cultural ideas, so here goes.
If your parents can accept that a chicken egg is fertilzed or not, then we can start with that. An egg has exactly 1/2 the genetic information it needs to become a chick. The eggs we eat are laid by hens that have no roosters! If roosters were around, then the eggs would see sperm containing the other 1/2 of the genetic information and this would result in a chick developing from the egg. Same for people: we need information from both parents to develop. This genetic information is also required to form the blood within us.
Genetic information or genes are carried on chromosomes. Sperm cells in males and egg cells in females have only half the information that every other cell in our bodies has, but that information is split in a special way -- we have pairs of everything. Humans have 23 pairs of chromosomes. Each chromosome has a partner chromosome with another copy of each gene. Thus, a chick or baby won't develop until there are two copies of each chromosome - two copies of every gene. For the most part, each gene encodes one protein. The proteins are the workhorses of your cells.
Now, lets look at blood groups as an example of the mixing of pairs of genes. You could be one of 4 major blood groups: A, B, O or AB. There is a hint here - the blood group AB is a combination of the two genes, A and B - equally. O is the absence or mutation of both an A gene or B gene. A is either AA or AO when we look at the genes. B is either BB or BO when we look at the genes.
To explain it a bit further - and I am going to use my family as the example. My mother is B and my father is A.
One brother is A, one is O and I am a B. If I had a 3rd brother, he could have been AB. How? To get this, my mother has to have gotten a B gene from one of her parents (my grand mom) and an O gene from the other (my grand dad). My father got an A gene from one parent and an O gene from the other (I don't know their blood types). My A brother inherited an A from my dad and an O from my mother, so this would fit your parent's view. The O brother inherited two O genes, one from each of my mom and dad, thus his is a blend of the two. I inherited my mom's B gene and my dad's O gene, thus I would be exactly opposite your parents idea and that other brother I don't have would have inherited Mom's B gene and Dad's A gene - thus he too would be a blend of both parents. While your parents may view this sceptically, and I can only prove this here with words, please look up examples of the ABO blood groups on the web and understand that part of my backgroud is actually someone who has been trained to provide you with compatible blood if you need a transfusion!
While the ABO blood groups are pretty clear cut, many other genes truly do provide you with a blending of characteristics. With a Middle Eastern background, you will know that dark hair and dark eyes are very very prominent. I'm a green-eyed blond. My father is blue eyed, dark brown and my mom was hazel-eyed, black. Where does the blond come from? The green eyes? Green and hazel are blends of colour and blue only seen if other colours are absent or is a recessive trait. Blond is a recessive trait as well - brown and black will mask the blond colour. Thus if I married a black haired women, our children would be black haired and my trait would not be seen! But two recessive genes are seen as blond if my child married another blond - they could have blond or black haired kids.
The recessive traits are the problem with marriage between family members - the further apart the relationship, the less likely that recessive genes will form the pair that a child will inherit. Thus while cousin unions are only 2% more likely to share a recessive gene, brothers and sisters are more likely to be 25%. The more that members of small group marry and have children among members of their group, even the liklyhood of cousins having a problem child will grow. Examples of this are 1) Tay-Sachs disease among eastern European Jewish descent. In the United States today, approximately one in every 27 Jews is a Tay-Sachs carrier.; 2) beta thalassemia - a major blood anemia disorder, is very common in Iran, Sardinia, Malta and Cyrus. I found one paper that suggests 70% of Sardinians carry this disease. When a bad recessive trait is that common in a group or culture, the chance that a baby has the disease is 25% - way more a risk than the 2%.
As you searched out this site for an answer, please have a look at other sites that deal with genetics. Two great ones that you can find through Google searches are The DNA Learning Centre and the Online Biology Book.