In the following applet, the parabola is defined by the 4 draggable blue points.
The red draggable points are the freely choosen points on the plane.
Folds are in green. Cyan lines are just indicative (not constructed).
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Choose any two points P1 and P2 (preferably both "inside" the parabola)
Fold P1 on P2 (fold1)
That fold intersects parabola in M1 and M2
Fold M1 on M2 (fold2)
The two folds intersect in M3, midpoint of M1M2
Choose any point P3 (also preferably "inside" parabola)
Fold the segment bisector of M1M2 (fold2) onto itself, by a fold going through P3
This fold3 is then parallel to M1M2. It intersects parabola in M4,M5
Fold M4 on M5 (fold4)
Fold4 and fold3 intersect in M6, midpoint of M4M5. M3M6 (not constructed) is parallel to axis.
Fold M6 on M3 (fold5). This fold is perpendicular to M3M6, hence perpendicular to axis.
This fold intersects parabola in M7,M8, symmetric from axis.
Fold M7 on M8 gives the axis of parabola (Oy).
Axis intersects parabola in apex M9 = O.
Fold axis onto itself by a fold going through M9 gives the tangent at apex (Ox).