The possibility for the Jacobi equation to admit, in some cases, general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed functions of a Darboux-Bäcklund transformation for the trigonometric Darboux-Pöschl-Teller potential. As a result, one-step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter λ are constructed. For each n value, the eigenstates of these extended potentials are associated with a novel family of λ-dependent polynomials, which are orthogonal on]-1,1[. © 2015 AIP Publishing LLC.