Unavoidable parallel minors of regular matroids

We prove that, for each positive integer kk, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k)M∗(K3,k), M(Wk)M(Wk), M(Kk)M(Kk), the cycle matroid of the graph obtained from K2,kK2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,kK3,k by adding a complete graph on the vertex class with three vertices.