  Given a strictly increasing sequence $\mathbf{t}$ with entries from
$[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with
$|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\le i\}|+|\{c\in
\mathbf{c}\mid c\le i\}|\ge i$ for all $i$ in $[n]$. We can think of
$\mathbf{t}$ as a list of spots already taken in a street with $n$ parking
spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car
attempts to park in the $c_i$-th spot and if not available then proceeds up the
street to find the next available spot, if any. A parking completion
corresponds to a set of preferences $\mathbf{c}$ where all cars park.
  We relate parking completions to enumerating restricted lattice paths and
give formulas for both the ordered and unordered variations of the problem by
use of a pair of operations termed \textbf{Join} and \textbf{Split}. Our
results give a new volume formula for most Pitman-Stanley polytopes, and
enumerate the signature parking functions of Ceballos and Gonz\'alez D'Le\'on.
