  A permutation of length $n$ is called a flattened partition if the leading
terms of maximal chains of ascents (called runs) are in increasing order. We
analogously define flattened parking functions: a subset of parking functions
for which the leading terms of maximal chains of weak ascents (also called
runs) are in weakly increasing order. For $n\leq 8$, where there are at most
four runs, we give data for the number of flattened parking functions, and it
remains an open problem to give formulas for their enumeration in general. We
then specialize to a subset of flattened parking functions that we call
$\mathcal{S}$-insertion flattened parking functions. These are obtained by
inserting all numbers of a multiset $ \mathcal{S}$ whose elements are in
$[n]=\{1,2,\ldots,n\}$, into a permutation of $[n]$ and checking that the
result is flattened. We provide bijections between $\mathcal{S}$-insertion
flattened parking functions and $\mathcal{S}'$-insertion flattened parking
functions, where $\mathcal{S}$ and $\mathcal{S}'$ have certain relations. We
then further specialize to the case $\mathcal{S}=\textbf{1}_r$, the multiset
with $r$ ones, and we establish a bijection between $\textbf{1}_r$-insertion
flattened parking functions and set partitions of $[n+r]$ with the first $r$
integers in different subsets.
