  We consider an expansion of Presburger arithmetic which allows multiplication
by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a
parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies
in $\mathbb{Z}^k$, and we examine the counting function $|S_\mathbf{t}|$ as a
function of $\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can
be expressed as an eventual quasi-polynomial (there is a period $m$ such that,
for sufficiently large $t$, the function is polynomial on each of the residue
classes mod $m$). We show that such a nice expression is impossible with 2 or
more parameters. Indeed (assuming \textbf{P} $\neq$ \textbf{NP}) we construct a
parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even
polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric
sets $S_\mathbf{t} \subseteq \mathbb{Z}^d$ with arbitrarily many parameters,
defined in a similar language without the ordering relation, we show that
$|S_\mathbf{t}|$ is always polynomial-time computable in the size of
$\mathbf{t}$, and in fact can be represented using the gcd and similar
functions.
