To explore the essential law of fuzziness of interval-valued intuitionistic fuzzy sets (IVIFSs), the entropy brought by interval values is studied. Moreover, together with classical score and accuracy functions, a series of novel parameterized generalized score functions on IVIFSs are proposed. First, the information that IVIFSs load is divided into three parts, where one part is the normalized score value, the second one is the classical accuracy value, and the third one is the newly proposed entropy value which originate from interval values. Second, by using the proposed entropy function and the other two parameter functions, a series of novel generalized score functions for IVIFSs are introduced. Furthermore, some properties on these proposed functions are explored. The main technological innovation of this paper is to express the relationship between the generalized score function and the three parameters introduced by using trigonometric functions. The main characteristic of this paper is that smaller the entropy value of IVIFN, the bigger is the generalized score function value. Finally, an example illustrates that the newly proposed generalized parameterized score functions deal with decision-making information in more detail, where a variety of decision-making plans can be provided for the decision makers. Therefore, the decision-making results obtained by the proposed generalized score functions are more expected.