In the above graphic, the point travels along a path on the swallowtail. This path is the set of points (a, b, c) such that p(x) = 0, p'(x) = 0, and p''(x) = 0 for the corresponding polynomials. In other words, this path represents the polynomials with a triple root. Regarding the regions of the swallowtail, this path forms the edge between the wings of the swallowtail (the area corresponding to polynomials with one double root) and the underside of the tail (the area corresponding to polynomials with one double root and two single roots). Therefore, it makes sense that the polynomials represented by this edge have a triple root. For a discussion of a similar cusp-curve in two dimensions, see Project One.
Notice that at the catastrophe point (the origin in parametric space), the corresponding polynomial is p(x) = x4, and its root is the quadruple root. This point is the only point corresponding to a polynomial with a quadruple root, because it is the only point for which p(x) = 0, p'(x) = 0, p''(x) = 0, and p'''(x) = 0.
To learn about the relationship between the polynomials represented by points on a plane tangent to the swallowtail, see Tangent Plane.