In the US system, interest groups have much more power over policy-making than parties do. We can see that in the example of iron triangles.
In an iron triangle, the political party really only has power over who gets to be on a given Congressional committee. Outside of that, the party has little control and the interest groups have a great deal.
Obviously, the interest group is one corner of the triangle. This gives it a great deal of power through its ability to give campaign contributions to members of Congress and through its ability to work closely with the government agency. The party does not have these means of influence. The interest group will work closely with the Congressional committee members and it will also try to influence the bureaucracy. It cares more about this area of policy than any other and puts all its resources toward it. By contrast, the party has many other things to be doing and does not have the same ability to exert pressure on the various parts of the iron triangle.