The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.
Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.
Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.
So no, you cannot make a parabola look like a branch of a typical hyperbola.
But you can make a branch of a hyperbola look almost like a parabola.
The match will still not be exact. You might as well ask for a positive number that is exactly zero.