It sounds to me that they just mean that each bound can be seen as a function of the parameter(s) in the parametrization and this function is Lipschitz continuous.
An example: Consider the XY-plane. Let $Y(x)$ be a function of $x$. This function can be seen as describing the upper bound of the area below the graph.
This function can then have the Lipschitz property, which means that for all $x_1$ and $x_2$, we have
$$
|Y(x_1)-Y(x_2)| < K|x_1 - x_2|
$$
for some constant $K>0$ independent of the $x_1$ and $x_2$ chosen.
If you are not so familiar with Lipschitz continuity, you can interpret this that the function is quite nice. It is continuous and even more.
You might for example note that all difference quotients are bounded. Take any $x$ and some small $h>0$, and choose $x_1 = x+h$ and $x_2=x$.
$$
|Y(x+h)-Y(x)| < Kh \Leftrightarrow \frac{|Y(x+h)-Y(x)|}{h} < K
$$
So if the function has a derivative in some point its absolute value is bounded by $K$. The absolute value of slope between two points on the graph is always bounded by $K$
A Lipschitz continuous function does not have to be differentiable though, but it can in some way be seen as being between continuous functions and differentiable functions on the "niceness" scale.
Any continously differentiable function is, at least locally (like in a bounded interval), Lipschitz. This is easy to prove from the definition.
You have the corresponding definition in several dimensions if you have functions of several variables and/or vector valued functions: If $Y$ is a function from the parameters where the output is vector of bounds, the Lipschitz condition is more or less the same but with vector norms used:
$$
|| Y(x_1) - Y(x_2) || < K ||x_1 - x_2||
$$
for all $x_1, x_2 \in \mathbb{R}^n$ and $Y(x_1), Y(x_2) \in \mathbb{R}^m$ for some dimensions $n$ and $m$.
