Does "isoparametric" mean what it sounds like? Please forgive my inexact language. Given an infinitely dense family of smooth curves on $\mathbb{R}^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an _isoparametric curve_. For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve. Is this a proper use of the term _isoparametric_?

Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.

Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.