Making a parabola from scratch is harder. Here's one way to find points on a parabola using a ruler, compass, and some graph paper.
The definition of a parabola is the set of all points that are the same distance between a point and line. We'll use this definition for our construction.
Take a piece of graph paper. Color one of the horizontal lines close to the bottom of the page blue. This will be our line from the definition. Now pick one of the verical lines close to the middle of the page, and find the point on the vertical line two units above the blue line. This will be our point from the definition, so color it blue, too. You're paper should look something like this.
Now we're going to start finding points on the parabola, so get out a different colored pencil, say red. The first point is easy -- it's just the point right below the blue point, one unit above the blue line. Color it red.
The other points are trickier, and now you need the compass. Open the compass so it spans the length of two squares from your graph paper. Put the point of the compass on the blue point, and draw a full circle. Find where the circle crosses the second line up from the blue line. There should be two crossing points. Color them red -- they are on the parabola.
Now you repeat what you've just done, replacing "two" and "second" by "three" and "third". In other words, open the compass to three units, and draw a circle around the blue point. Find where this circle crosses the third line up from the blue line, and color those two crossing points red.
Repeat all this for four, five, six, and so on, until you are off the page. Do you see how the construction of these points is following the definition of the parabola? Connect the dots as smoothly as you can to make the shape of the parabola.