This is a short section to introduce a very broad topic. Parametric equations are a maths construct that permit us to graph equations which are not functions. Granted, that we will frequently and incorrectly refer to them as functions, but technically, they are relations. As the adjective tries to explain, parametric equations have one or more parameters and it is these parameters that get varied in order to permit the computation of graphical elements, such as $x$,$y$, and $z$. The usual example of a “non-function” parametric equation is the circle with its parameterization $x=\cos(t)$ and $y=\sin(t)$.
One of the annoyances in understanding functions is that to graph them it is usually needed to iterate through some variable, for example $x$, and obtain some other variable, for example $f(x)=y$. In the world of third grade math, a line is $f(x)=mx+b$ and by iterating some domain of $x$ we derive matching values for $f(x)$ which when graphed yield a straight line. These types of functions assure that there is one and only one value of $f(x)$ for every value of $x$. Other types of relations exist, in which the function has more than one argument. Consider $y^{2}+x^{2}=r^{2}$. You will recognize this as the function for a circle of radius $r$. We call this an implicit function because it is not yet solved for one of the two variables. Furthermore, the values which exist are confined to a certain set of $x$ and $y$ values and the relation is not defined over all $x$ and $y$. Conic sections and their equations are often left as implicit equations because in that form they can be easily written and understood, whereas solving them for one of the variables results in one or more radicals and unwanted complexity.
Somewhat less common in real life, but plentiful otherwise are functions of just one variable that cannot be solved for that variable. A good example of an implicit function in one variable is $y=xe^{x}$; solve for $x$. Using Newton's method, we can certainly find any numerical solution. In this context, a solution is an $x$ value that causes $y$ to be zero. We can also make a graph of the function. However, an analytic solution of this implicit function is well above the third grade level.
Not long ago, in human years, Computer Algebra Systems (CAS) would only graph explicit functions. More recently, we have seen CAS that will handle implicit functions. That is a logical advancement, since to graph an implicit function is not difficult but is certainly tedious. So let computers do what they do well.