I believed that giving blood and being on the donor register was doing a good act but my dad thinks this is haram. Does anyone have any advice?
Your perspective would be more correct in compare with your respected father's view. In truth, donating blood can be considered as a great profitable act which can save the life of many individuals. But it can be counted not as an appropriate practice if it make you a remarkable harm. In other word, it can be considered as an impermissible practice if your donating blood made you a significant harm for your body. But in common form, and actually in majority of the times it can be counted as a helpful act and might be it does not made a remarkable harm for majority of people. As a consequence, it ought not to be seen as a negative issue.
And Allah knows best