Do sparse graphs have to be large?
The common definition is that a family of graphs $G = (V, E)$ is sparse if $m \in o(n^2)$, with $n = |V|$ and $m = |E|$. Formally, that requires the family to be infinite, and we only know something in the limit.

The definition does not apply to graphs of a fixed size per se. One would certainly want $m \ll n^2$ in some sense. For instance, if $m \leq 5n$ or $m \approx \log n$ you would probably say that your graph is "sparse".

You can bridge this gap by requiring a bit more on the asymptotic side. Say we call a family of graphs sparse if

$\qquad\displaystyle m \leq f(n) + O(1)$ with $f \in o(n^2)$;

then we may want to call every graph with $m \leq f(n)$ sparse. The implicit assumption is that the error term hidden in $O(1)$ is well-behaved.