  We determine local topological types of binary differential equations of
asymptotic curves at parabolic and flat umbilical points for generic
$2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective
classification of Monge forms and classification of general BDE obtained by
Tari and Oliver. In particular, generic bifurcations of the parabolic curve are
classified. The flecnodal curve is also examined by direct computations, and we
present new bifurcation diagrams in typical examples.
