Given a drawing of a parabola is there any geometric construction one can make to find its focus? This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci?
I think this is harder, I can't even find is axis or vertex.

As @AchilleHui mentions, the midpoints of two parallel chords lead to the point of tangency ($T$) with a third parallel line. Note that the line of midpoints is parallel to the axis of the parabola. By the reflection property of conics, the line of midpoints and the line $\overleftrightarrow{TF}$ make congruent angles with the tangent line. 

So, two sets of parallel chords determine two points of tangency and two lines-of-midpoints, which determine two lines that meet at focus $F$. $\square$
