Parametric equations are equations in terms of a single parameter. An example is the parametric equation for a circle: x = r cos(t), y = r sin(t), for a circle with radius r with rectilinear coordinates x and y. It’s best if the parameter represents a causal variable but time is often used whether or not time causes anything.
One advantage of parametric equations is that the arc length is also a function of the parameter so the parameter may be considered as a position on the curve. This has been seductive with time, so much that time is thought of parametrically as if it were a one-dimensional variable. To see the multiple dimensions of time one must see it in terms of coordinates.
We could parameterize space and de-parametrize time to switch the places of the two. For example, on a road trip the odometer measures the trip length, which would be the parameter. If the travel time in the E-W and N-S directions is measured with the trip length, the trip may be plotted as two dimensions of time and one dimension of space. Before and after would be associated with smaller and larger values of the odometer.
If the fuel gauge were precise enough, the fuel level could be used as the parameter for both travel time and distance. Because of causal connections, it may well make sense to treat some variable parametrically, but either time, space, or some other variable can serve the purpose. Much of the character of time comes from its parametric association.