A parking function is a function $π:[n]\to [n]$ whose $i$th-smallest output is at most $i,$ corresponding to a parking procedure for $n$ cars on a one-way street. We refine this concept by introducing preference-restricted parking functions, which are parking functions with codomain restricted to some $S\subseteq[n]$. Particular choices of $S$ yield new combinatorial interpretations of previous results about variant parking procedures, and new results too. In particular we consider prime parking functions, parking procedures with fewer spots than cars, and parking functions where each spot has space for multiple cars. We also use restricted parking functions to reprove Abel's binomial theorem.