The way the first sentence is written, it isn’t saying this happened when you were a child, although that is what the writer meant. (Bet you didn’t know that mindreading is a necessary skill for editors, did you?) Since this didn’t happen when your parents were a child (two parents were one child? WTF?), that opening needs to be rewritten.
Also note that doctor’s has an apostrophe in the second sentence.
When you were a child, your parents kept you from seeing a doctor, even when you were sick. As an adult, you decide to go to your first doctor’s appointment. As the doctor draws your blood, he is horrified to discover that it is green.
This would not happen. Part of the reason humans look pinkish/ruddy (even humans with dark skin) is that we have red blood. Green blood would cause the character to have a greenish cast (if not outright green color, if there’s not enough melanin to mask it) to their skin. The fine blood vessels in his eyes would show green. The inside of his mouth would be greenish, as would all other parts of his body that lacked or had little other pigment. There is no way he could reach adulthood without such a difference being apparent even without a blood test.
Did this character never skin his knee as a child? Never get a nosebleed? Never lose a tooth?
Also, it’s likely that the dietary requirements of a mammal with green (copper-based, most likely) blood would be quite different from those of mammals with red (iron-based) blood. At the least, he’d need to take supplements as well as eat a lot of legumes and tree nuts to avoid a serious deficiency.
(If I can see so many holes in the concept without even trying, chances are that someone else would notice, too.)
As a variation on the ‘you find out your parents were keeping the secret that you’re an alien’ trope goes, this one shows a serious lack of thinking through the basic science.