  A smooth ruled surface in 4-space has only parabolic points or inflection
points of real type. We show, by means of contact with transverse planes, that
at a parabolic point, there exist two tangent directions determining two planes
along which the parallel projection exhibits $\mathcal A$-singularities of type
butterfly or worse. In particular, such parabolic point can be classified as
butterfly hyperbolic, parabolic, or elliptic point depending on the value of
the discriminant of a binary differential equation (BDE). Also, whenever such
discriminant is positive, we ensure that the integral curves of these
directions form a pair of foliations on the ruled surface. Moreover, the set of
points that nullify the discriminant is a regular curve transverse to the
regular curve formed by inflection points of real type. Finally, using a
particular projective transformation, we obtain a simple parametrization of the
ruled surface such that the moduli of its 5-jet identify a butterfly
hyperbolic/parabolic/elliptic point, as well as we get the stable
configurations of the solutions of BDE in the discriminant curve.
