A parabolic representation of a knot  is a $PSL(2,C)$-representation which sends a meridian to a parabolic element. In 70's,  R. Riley studied these representations of two bridge knots and obtain a complete classification via a single polynomial $R(y)in mathbb{Z}[y]$. We generalize the Riley's polynomial to any knot diagram with a base crossing. By utilizing parabolic quandle theory  with the aid of Groebner elimination software, we can obtain a complete classification of parabolic representations for thousands of knots, including all knots up to 12 crossings.  In particular, based on these computational data, we can easily obtain hyperbolic volume and Chern-Simons invariants by an explicit diagrammatic formula without using SnapPEA software. If time permits, a concrete  example computing a generalized Riley polynomial will be presented.