Even parking functions and spanning trees of complete bipartite graphs Set $\mathbb{N} := \{0,1,2,\ldots\}$. A parking function of length $n$ is a sequence $(\alpha_1,\ldots,\alpha_n) \in \mathbb{N}^n$ whose weakly increasing rearrangement $\alpha_{i_1} \leq \alpha_{i_2} \leq \cdots \leq \alpha_{i_n}$ satisfies $\alpha_{i_j} \leq j-1$ for all $1 \leq j \leq n$. Let $\mathrm{PF}(n)$ denote the set of parking functions of length $n$. Let $K_{n}$ denote the complete graph on $n$ vertices. It is very well-known that
$$\#\mathrm{PF}(n) = \#\textrm{ of spanning trees of }K_{n+1}=(n+1)^{n-1}.$$
(This sequence is http://oeis.org/A000272.) Moreover, there are many explicit bijections between parking functions and labeled trees, with interesting properties. I am interested in a bipartite analog of this result. Let $K_{m,m}$ denote the complete bipartite graph with $m$ vertices in one part and $m$ in the other part. It is not hard to show that
$$ \#\{(\alpha_1,\ldots,\alpha_{2m-1}) \in \mathrm{PF}(2m-1)\colon \alpha_i \textrm{ is even for all }i\} = \#\textrm{ of spanning trees of }K_{m,m} = m^{2m-2}.$$
(This sequence is http://oeis.org/A068087.) I wonder if there is a simple bijective proof of this last equality. This problem is a special case of (3) in Section 5 of http://arxiv.org/abs/1506.03470.
 A: This is far from an answer, but only a possible first part of such a bijection. (There were two plainly wrong bijection parts in the first version that cannot be fixed.)
The bijection between factorizations of the long cycle $(1,\ldots,2m)$ and parking functions of length $2m-1$ described in my answer https://mathoverflow.net/a/94241/21291 restricts to a bijection with the desired property:
First: To fit the context of the other answer, I use $1 \leq \alpha_{i_j} \leq j$ rather than your $0 \leq \alpha_{i_j} < j$ and consider thus parking functions of length $2m-1$ with all parts being odd.
Lemma: A parking function $p = p_1\cdots p_{2m-1}$ has all parts odd if and only if the factorization $\Psi^{-1}(p) = (i_1j_1)\cdots(i_{2m-1}j_{2m-1})$ has the property that all first entries $i_1,\ldots,i_{2m-1}$ are odd.
Corollary: The bijection $\Psi$ restricts to a bijection between factorizations of the long cycle $(1,\ldots,2m)$ into transpositions $(i_kj_k)$ with all $i_k$ odd and parking functions of length $2m-1$ with all parts being odd.
Using this observation it would be thus left to find a bijection between such factorizations and spanning trees of $K_{m,m}$. 
A: Let $n=2m$ and let $(\alpha_1,\dots,\alpha_{2m-1})$ be an even parking function.  Apply Pollak's map: $c=(\alpha_2-\alpha_1,\dots,\alpha_{2m-1}-\alpha_{2m-2})\mod 2m$.
Thus $c$ is an $(2m-2)$-tuple of even numbers between $0$ and $2m-1$.
Let $a=(c_1/2,\dots,c_{m-1}/2)$ and let $b=(c_m/2+m,\dots,c_{2m-2}/2+m)$.
Then $(a,b)$ is the 'bipartite' Prüfer code of a spanning tree of $K_{m,m}$, where the vertices of one block are labelled $0,\ldots,m-1$ and the vertices of the other block are labelled $m,\ldots,2m-1$.
