Take two oppositely charged rods, place them separately near small pieces of paper. Why do they both attract small pieces of paper? Is there any third type of charge on papers which attracts both positive and negative charges?

The pieces of paper are initially neutral. When a charged rod is brought near, it induces a charge separation in the paper—one side becomes slightly positive and the other slightly negative. Since opposite charges attract, the paper gets attracted to both positively and negatively charged rods. There is no third type of charge; it’s just an induced charge effect.