The relationship between a curve and its derivative is mutual, and looking at it the other way round will explain why the derivative of a parabola is a straight line: Because the integral of a straight line is a parabola.
Others have mentioned what the intuitive meaning of a derivative is: How steep the curve is at that point, i.e. how steep a line would be that touches the curve only at that point: a tangent.
So why is the derivative of a parabola a straight line? The first thing to note is how the derivative line crosses the x axis precisely where the slope of the parabola is horizontal, i.e. its "steepness" is 0. Before that the derivative was negative. This means that the parabola curve falls with increasing x, which is exactly what we see. Afterwards the slope is positive; the y value of the parabola grows with growing x.
So we have established that the derivative line is qualitatively correct, but it could still be a curved line. Why is it straight? This picture may explain it:
We see a parabola in red and its derivative in green. Because the reversal of deriving is integrating, the parabola gives us the value for the area under the green curve for each x. The areas are triangles which I highlighted in shades of yellow and orange. Remember that the area of a triangle is half the area of a corresponding rectangle, marked in a very light shade above the green line. Therefore, the triangles' area — which is the area under the green line — is growing with the square of x! A function whose value grows with the square of its argument is a parabola. I have marked the triangles' areas and the corresponding value of the parabola with arrows. It's actually easy to just count the areas with the aid of the grid.
Soooo — if integrating a straight line results in a parabola, deriving a parabola must result in a straight line!
A nice classical real world application of such a relationship is speed and distance during steady acceleration. The speed indicates how fast we move, and would be the green line in my image. The distance covered so far is the red parabola. We are accelerating, so the speed grows over time (the green line points up). "Steady acceleration" means here that the speed graph is a straight line.
If we have the speed of a vehicle at each point after it started, we can compute the distance covered: It is the area under the speed graph, in our diagram the triangles. The red graph shows that distance. It becomes steeper and steeper, because the vehicle accelerates. It is a parabola because the triangles' area grows quadratically.
A nice example with more graphics for the speed/distance relationship can be found here.