Even parking functions and spanning trees of complete bipartite graphs

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 $K_{m,m}$.