$n-1$

Replace the points $(x,y)$ by $(x,y-x^2)$. What happens?

We do the replacement as suggested in the hint. This replaces the parabolas between the old points by the lines between the new points. So we get the same problem as before, but now with parabolas replaced by straight lines everywhere, and this is of course much simpler: If the convex hull of the $n$ points is a $k$-gon, then obviously at most the $k$ boundary lines are good, since every other line has points on both sides of it or a third point on it. But among the $k$ boundary lines, at most $k-1$ of them are good, since at least one of them has the convex hull above it instead of below (take the line containing the lowest point of the convex hull). Since $k \le n$, we therefore have at most $n-1$ good lines, and hence at most $n-1$ good parabolas. Clearly, this is sharp as one can see from just choosing the points on the $x$-axis (before the change of variables).