In alpha and beta decay the parent element and the daughter element can be

In alpha and beta decay the parent element and the daughter element will be different isotope and not the same. 
This is because as alpha particles comes out; the new element will have two less in atomic number; that is the atomic number will reduce by two and the mass number will reduce by four. While in a beta particle decay, the new element will have one higher in atomic number; that is the atomic number will increase by one.