Exceptional polynomials are complete orthogonal polynomial systems with respect to a positive measure in the real line which in addition are eigenfunctions of a second order differential operator. The most apparent difference between classical orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials (as it happens with the classical families of Hermite, Laguerre and Jacobi) although they form a complete orthonormal set of the underlying \(L^2\) space defined by the orthogonalizing positive measure. Since 2015, some examples of exceptional Jacobi polynomials depending of continuous parameters (apart from the parameters associated to the Jacobi family) have been constructed. The more general construction appeared in [the author, Stud. Appl. Math. 148, No. 2, 606--650 (2022; Zbl 1533.33012)]. In fact, there is computational evidence that shows that all the other examples are particular cases of those introduced in [loc. cit.], although it remains as an open problem to prove it. The families of exceptional Jacobi polynomials depending on continuous parameters are constructed using rather different methods. The purpose of this paper is to prove that for the examples constructed in [loc. cit.] the situation is different. The authors shows that the exceptional Jacobi polynomials can be divided in two classes: in one of these classes the families are indeed deformations of Jacobi polynomials, but the families in the other class are deformations of standard families of exceptional Jacobi polynomials.