I know this seems like an incredibly general or even silly/random question, however, please allow me to explain why I'm asking this question.
Someone who, for example, didn't understand that trigonometric functions and circles are related, would be missing out on a whole genre of important possible insights about trigonometry and circles. I feel that I am missing a similar important understanding of a connection between parabolas and circles.
Note that if you plot a pendulum's position against time it would form a sine/cosine function. The force that sets a pendulum in motion is gravity. That same force causes a ball to fall in a parabolic path.
Circles -> sine functions -> pendulum -> gravity -> parabola.
Something else led me to this question, but this was my way of explaining it. I feel like I'm missing an insight that is deep, beautiful, and important.