A canonical bijection from linear independent vectors to parking functions

They are in canonical bijection with the spanning trees of the complete graph $K_{n+1}$ (for which the bijections with parking functions are well known).

Indeed, let $K_{n+1}$ be the complete graph on the ground set $\{0,1,\ldots,n\}$. Denote $f_0=0$ and consider $n$ linearly independent vectors $f_1,\ldots,f_n$. Denote further $e_j=f_j-f_{j-1}$ for $j=1,\ldots,n$. They form another basis of the same $n$-dimensional space $W$ as $f_j$'s. For an edge $\epsilon=ij$, $i<j$, of $K_{n+1}$ we consider the vector $w(\epsilon)=f_j-f_i=e_{i+1}+\ldots+e_j$. Note that $n$ edges $w(\epsilon_1),\ldots,w(\epsilon_k)$ are linearly independent if and only if the set of edges $\epsilon_i$'s does not contain cycles. Thus the bases of $W$ correspond to spanning trees of $K_{n+1}$.

The above construction is a standard vector representation of the circuit matroid.

See "A braid group action on parking functions" by Gorsky and Gorsky.