The two branches of the parallel curve a distance away from a parametrically represented curve are
References
Gray, A. ``Parallel Curves.'' §5.7 in
Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 95-97, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 42-43, 1972.
Lee, X. ``Parallel.''
http://www.best.com/~xah/SpecialPlaneCurves_dir/Parallel_dir/parallel.html.
Yates, R. C. ``Parallel Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 155-159, 1952.