What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?
You can take any degree 3 bipartite graph $G$ and take its disjoint union $G'$ with a cycle $C$ of length 2m. The new graph $G'$ is bipartite, and has average degree $\frac{3n + 2m}{m+n} = 2 + \frac{n}{n+m}$. Also, the number of perfect matchings in $G'$ is exactly twice the number of perfect matchings in $G$, because the perfect matchings of $G'$ are the disjoint unions of a perfect matching of $G$ and a perfect matching of $C$, of which there are two. So, there is a polytime parsimonious reduction from counting perfect matchings in 3-regular bipartite graphs to counting perfect matchings in graphs with maximum degree 3 and average degree $2 + \frac{1}{\mathrm{poly}(n)}$.