Any way to center text within a \parbox?

just use \centering inside the box.

\parbox{2cm}{\centering
  one two three four five six}

I assume you are using this package, so with some limitations you can do this:

\documentclass{article}

\usepackage{pbox}

\begin{document}

\parbox{2cm}{\centering
  one two three four five six eight nine ten
aaaaaa\\
bbb}

\bigskip

\pbox{5cm}{\relax\ifvmode\centering\fi
  one two three four five six eight nine ten\\
aaaaaa\\
bbb}


\end{document}

There is the varwidth environment from the package with the same name, which is an auto-sizing version of minipage. You only need to give a maximal width.

The text centering inside it or inside \parbox can and should be done using \centering.