  Naples parking functions were introduced as a generalization of classical
parking functions, in which cars are allowed to park backwards, by checking up
to a fixed number of previous spots, before proceeding forward as usual. In
this work we introduce the notion of a complete parking preference, through
which we are able to give some information on the combinatorics of Naples
parking functions. Roughly speaking, a complete parking preference is a parking
preference such that, for any index $j$, there are more cars with preference at
least $j$ than spots available from $j$ onward. We provide a characterization
of Naples parking functions in terms of certain complete subsequences of them.
As a consequence of this result we derive a characterization of
permutation-invariant Naples parking functions which turns out to be equivalent
to the one given by (Carvalho et al., 2021), but using a totally different
approach (and language).
