Can you get HIV from touching a rag/paper towel/ tissue with period blood on it?

Yes but.... Any body fluids should be considered a biohazard and theoretically can transmit infectious diseases (such as hiv) if that person is hiv+. Although the risk is small, using proper precautions (latex gloves, etc.) will eliminate that risk.
Use gloves. Transfer of HIV has to occur from blood to either a mucous membrane (eye, mouth, penis, vagina, rectum) or enter through a cut/scrape/microsopic break in the skin. Yet, if you see blood on anything from someone else, don't touch it without wearing gloves whenever possible as a "universal precaution".