How could I improve my explanation of parameters and parametric equations?

I think you have lots of room for improvement. I recommend you begin with excerpts from the OpenStax calculus textbook. Specifically,

• this section from volume 2 about parametric equations:

https://openstax.org/books/calculus-volume-2/pages/7-1-parametric-equations

• and the first few pages of this section from volume 3 about parametric surfaces:

https://openstax.org/books/calculus-volume-3/pages/6-6-surface-integrals

You could think of the difference between "parameters" and "variables" this way:

A parameter is an independent variable used to restrict coordinate variables such as $(x,y,z)$. For example, we might have the coordinate variables depend on $t$ by $x=x(t),\ y=t(t),\ z=z(t)$. This will typically result in a curve. Or we might have the coordinate variables depend on two parameters $(u,v)$ by $x=x(u,v),\ y=y(u,v),\ z=x(u,v)$. This will typically result in a surface.

The parameter can be abstract, disembodied from the $(x,y,z)$ space. For example, $t$ might be thought of as "time". Or the parameter can be more concrete. For example, when we talk about the graph of a function $y=f(x)$, the parameter is $x$. In other words, $x=t, y=f(t)$. Another typical parameter for a curve is the "arc length" parameter $s$ which is the distance along the curve from a specific point to a "moving" point $(x(s),y(s),z(s))$.

Intuition is built by trying to interpret abstract parameterizations. Here is an example:

$$[x(u,v)=\cos(v),\ y(u,v)=\sin(v),\ z(u,v)=u]$$ After we think about this and draw some pictures, we realize that we have parametric equations for a cylinder along the $z$-axis. So the $v$ parameter is an angle variable, and the $u$ parameter is the axial variable.

A reason why parametric curves and surfaces enter into calculus is because we need calculus to quantify the "distortion" of the transformation from the parameter domain to our $(x,y,z)$ spatial domain. For example, if we have a parametric curve in the plane given by $x=x(t),\ y=y(t)$, then the increment of length $ds$ along the curve is given by $ds=c\cdot dt$, where the distortion factor is $$c=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$

For parametric surfaces, the "area distortion" factor $J$ is given by equation (6.18):

https://openstax.org/books/calculus-volume-3/pages/6-6-surface-integrals

When you unravel this, it involves the six partial derivatives of the coordinate variables $(x,y,z)$ with respect to the parameters $(u,v)$:

$$\frac{\partial x}{\partial u},\frac{\partial x}{\partial v},\frac{\partial y}{\partial u},\frac{\partial y}{\partial v}, \frac{\partial z}{\partial u},\frac{\partial z}{\partial v}$$

There are multiple senses to the term 'parameter', but as a general principle, parameters are types of variables used to selectively constrain another set of variables through a functional dependency.

Example 1: Parameters to Constrain Sets

In this case, a parameter is a variable which is generally held constant within a specific context where other variables are allowed free to vary. This means that we have to draw distinctions when terms like 'variable' and 'constant' are used. It might help to make sense of it through a concrete example.

• Pi is a constant. It, in an abstract sense, represents a value that is presumed to be real and independent of any particular decimal representation.
• The coordinates x and y on the Cartesian plane are variables when are generally allowed to vary in each and every situation. For instance, in the general linear form ax + by = c have a distinct (a,b,c) for all (x,y). Note that we usually call a, b, and c constants and x and y variables.
• Strictly speaking, however, we are free to vary those "constants" to express different lines. This makes those "variable constants" parameters. By using the term parameter, we are indicating that technically speaking those symbols are neither variable nor constant strictly speaking.

Thus, if we talk about a unique line, then a, b, and c do not vary and are called parameters, but if we talk about a unique plane, then they do vary from distinct line to line. What the term parameter indicates then, is how we bind the variable to a particular domain of discourse. Thus:

• Strictly speaking a constant like Pi or e never varies as a number.
• Strictly speaking a variable is allowed to always vary across a domain of discourse.
• And strictly speaking, a parameter is treated like a constant in some contexts and a variable in other contexts where contexts are delineated by the binding of the symbols under different domains of discourse.

This first example shows then how parameters as additional variables enriches the description of Euclidean lines by allowing us to hold them constant to select a specific set of lines on a plane (including a set of a single line or an empty set). The triple (a,b,c) refers to the domain of discourse regarding which line or lines on the plane if any.

Example 2: Parameters as Additional Degrees of Freedoms

In parametric equations, additional domains of discourse can be introduced in order to add additional dimensions of analysis and enrich our model. For instance. A clear example of this is provided in user52817's answer. Here, consider that in a volumetric space, it might be handy to expand the discourse to include a time variable above and beyond the three coordinates of the space. Thus, the triple (x,y,z) is enriched to (x(t),y(t),z(t)). Now, it is possible in a rudimentary anticipation of spacetime to discuss the model as four-dimensional.

Here, the parameter again constrains the original variable by allowing the model to discuss a volume at time t such that (x,y,z) ∀x,y,z∈R ∃!t∈R. Thus, as a rudimentary Euclidean model of physical space permits volumetric slices at each t in the domain of discourse. Again, like the linear equation example, introducing this parameter allows the model to treat the parameter both as an independent variable (to accommodate the flow of time) and a constant (to look at points of a particular space at any given instant).

This second example shows then how parameters as additional variables enriches the description of Euclidean space by allowing us to hold a constant time to select a specific subset of volumes given by a set t (including a set of a single instant or an empty set). The singleton (t) refers to the domain of discourse regarding which instant or instants on the timeline if any.

Final Note on Dependent and Independent Variables

In both examples, notice that (x,y) can be said to depend on (a,b,c) and (x,y,z) can be said to depend on (t). But such dependencies are often aligned to the use of the model. Remember that in a functional dependency, particularly if the domain and co-domain are described by a one-to-one relationship, then the choice to hold one variable as dependent on another variable is a matter of preference or convention. Let's exemplify.

An easy example is the linear function which by convention holds the ordinate as dependent on the abscissa. For any line y(x) it is possible to express the same line in the form x(y). Thus, while y(x) = mx + b makes finding the y-intercept simple, through algebraic manipulation of the same line, we can express the line as x(y) = m'y + a where the x-intercept is easier to see. So, in any discussion of one set of variables depending on another, dictating which terms are the "parameters" tends to prescribe which variables are dependent on others.