What is a the intuition behind a parametric equation?

A good way to look at it is this ...

Parametric equations let you draw curves with an Etch-a-Sketch.

(Image credit: pikajane @ deviantart )

The above illustration is a bit extreme, but in theory you could re-create the image by coordinating the toy's knob-turns in just the right way. This is what parametric equations help you do.

Imagine giving control of each Etch-a-Sketch knob to a separate person, Lars Rue for Left-Right, and Ursula Dobbs for Up-Down. And imagine tasking LR and UD with drawing a perfect circle. Chances are, this wouldn't go well at first ---or maybe at all-- with each kid anticipating and/or reacting to the other's movements. There'd be lots of squiggles and back-tracks ... and perhaps more than a few colorful exclamations.

Now imagine setting a metronome nearby. Tick ... tock ... tack ... tuck .... With any luck LR and UD will stop paying that much attention to each other, and just pay attention to the metronome. After a little strategizing, the kids might decide to start their circle at "$(1,0)$" and proceed thusly:

• tick : LR turns his knob leftward, and UR turns hers upward.
• tock : LR keeps going left, and UR goes down.
• tack : LR reverses, going right, while UD keeps moving downward.
• tuck : LR keeps right, and UD switches to up.
• Repeat.

With some practice, LR and UD learn better how to far to move their knobs, how to ease into and out of each change of direction, settling into nice, fluid motions that are essentially the same for each kid, but "out of step" by one tick of the metronome. The better they get, and the more-precisely the circle gets traced, the more the kids find that their rhythmic left-rights and up-downs exactly mimic the rise-falls of the (co)sine wave.

Importantly: Once the kids get really good at tracing the circle, they don't have to look at it ---or each other--- any more. Just listening to the tick ... tock ... tack ... tuck ... of the metronome is all they need. Their motions are governed by time.

But notice: time is not part of the drawing itself. It doesn't measure horizontal distance, or vertical distance. It's very much an "other", so its role is as a "para-meter" (literally, "beside measurement", or better here "other measurement").

By introducing time, LR (controlling the $x$ coordinate of a point) and UD (controlling the $y$ coordinate) are able to work completely independently to draw their circle. That's what parametric equations do: replace the relationship between $x$ and $y$ with dual relationships "$x$ at time $t$" and "$y$ at time $t$". Being able to deal with coordinates independently has plenty of advantages. (In physics, we notice that a cannonball's flight path is a parabola, because its up-down-ness is affected by gravity, whereas its left-right-ness is not.)

It's worth noting that the "good way to look at it" can be shortened:

Parametric equations let you draw curves.

The circle equation $x^2+y^2=25$ expresses a relation among the $x$ and $y$ coordinates of any point on the curve. The point $(0,5)$ is on the curve because $0^2 + 5^2 = 25$. The point $(-3, 4)$ is on the curve because $(-3)^2+4^2=25$. A computer (or Function Monkey) might plot the circle by taking $x$-values in sequence from left-to-right, computing the corresponding $y$-values (getting two at a time, except at $x=\pm 5$), but this ignores the inherent circle-ness of the circle.

Parametric equations, on the other hand, turn a static picture of points satisfying some Pythagorean relation, into a dynamic path that does loop-de-loops around the the circle. This has advantages, too ... not the least of which is that dynamic paths are just more fun.

Consider the Cardioid, which is drawn (in red) like this ...

(Image credit: Wojciech Swiderski via Wikipedia)

... according to these parametric equations: $$\begin{align} x &= 2 \cos t - \cos 2 t \\ y &= 2 \sin t - \sin 2 t \end{align}$$

The cardioid curve happens to contain all (and only) the points satisfying this relation: $$\left( x^2 + y^2 - 1 \right)^2 = 4 \left((x-1)^2 + y^2 \right)$$ but really ... how helpful is that to understanding the curve?

Whoops ... This answer seems to have turned into a big wall of text. I'll stop now. :)

A set of parametric equations is called by that name because they have a parameter. Such equations are often used when a graph might not have certain "typical" and/or "nice" properties like being one-to-one or onto, e.g., a circle.

For a circle, a set of parametric equations might look like this:

$$y=a\sin t$$ $$x=a\cos t$$

Then the circle of radius $a$ (centered at the origin) is generated by varying the value of $t$ and applying that value within each of the equations.