In this paper we describe moving frames and differential invariants for curves in two different $\left|1\right|$-graded parabolic manifolds $G/H$, $G=O(p+1,q+1)$ and $G=O(2m,2m)$, and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in $G/H$ inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in $G/H$ can be reduced...