There are a few possible directions to take with this situation, so I'll suggest a few and you can see what might suit you, and then we can help you figure out how to go about it. Okay?
Would you like to get more comfortable with touching menstrual blood? What is it about bloody fingers that you don't like or that's a turn-off for you?
Would you like any suggestions for how to get tampons without your parents' involvement? You do have the right to use whichever sanitary products you want to use, as it's your body.
Would you like any help with talking to your parents about tampons? Have they said what their objections are?
The kyriarchy usually assumes that I am the kind of woman of whom it would approve. I have a peculiar kind of fun showing it just how much I am not.