We consider the local geometry of a generic 1-parameter family of smooth curves in the real plane for which one member of the family has parallel tangents at two inflexion points. We study the equidistants of this family, that is the loci of points at a fixed ratio along chords joining points with parallel tangents, as a 2-parameter family depending on the value of the fixed ratio and on the parameter in the family of curves. Codimension 2 singularities of type ‘gull’ arise in this way and are in general versally unfolded by the two parameters. We also calculate the family of duals of the equidistants; here it is necessary to view them as bifurcation sets of bigerms and they evolve through ‘moth’ and ‘nib’ singularities also encountered in 1-parameter families of symmetry sets in the plane. Finally we show that certain sub-families of the 2-parameter family of equidistants can be classified by reduction to a normal form.