Biharmonic curves in a manifold are (real) curves that minimize the \(L^2\)-norm of the tension field. Since the tension field of a geodesic in parametrization by arclength vanishes, geodesics are biharmonic curves. But there can be biharmonic curves which are nongeodesic, and the paper is concerned with nongeodesic biharmonic curves in surfaces. Two basic facts are remarked about biharmonic curves. Along a nongeodesic biharmonic curve the Gauss curvature is constant and equal to the square of the geodesic curvature. As a consequence, if the Gauss curvature of the surface is nonpositive, the biharmonic curves are exactly the geodesics. Now the special case of biharmonic curves to surfaces of revolution is considered. If the Gauss curvature of the surface is nowhere constant, all biharmonic curves must be parallels. In the other direction, the surfaces of revolution are determined for which all parallels are biharmonic curves. Some examples of biharmonic parallels in surfaces of revolution with nonconstant Gauss curvature are also given. Finally, for surfaces of revolution with constant Gauss curvature, the biharmonic curve equation is solved explicitly.