Ink can sometimes get into a persons blood stream and cause ink poisoning through the blood.
But it would take a lot more ink than just simply writing on your hands or other parts of your skin for there to be enough ink to get into your blood stream to cause any health issues.
Such as soaking your hand in ink or pouring ink all over your hands or other parts of your body and then letting your skin soak the ink up.
But just writing on your skin with a pen or other ink is not gonna get into your blood stream and cause ink poisoning.
I've used ink pens to write on my skin and hands ever since I was 3 years old and it's never been an issue.