Abstract: In this paper I study composition operators induced by linear fractional selfmaps of the unit disc that are parabolic, but not onto. I show that such maps induce composition operators on the Hardy space H^p (1 <= p < infinity) that are decomposable. This result, the spectral properties of the operators being studied, and a recent result of Len and Vivien Miller, shows that such composition operators are not supercyclic. This work generalizes previous work of Gallardo and Montes, who proved the non-supercyclicity result, by different methods, for p=2, and it complements work of Robert Smith who proved decomposability for composition operators induced by parabolic automorphisms ("onto parabolics"). The paper features a significant expository component that makes it mostly self-contained.