Parabolic Dijkgraaf-Witten invariants of links in the $3$-sphere

We define a new invariant of links in the $3$-sphere and call it the parabolic Dijkgraaf-Witten (DW) invariant. This invariant is a generalization of the reduced DW invariant derived by Karuo. In this paper, we compute the invariant of several links over which double branched coverings are homeomorphic to the lens spaces. Moreover, we introduce a procedure for computing partial information of the parabolic DW invariant using only link diagrams.