How do I control parskip within a tcolorbox?

Yes, there is a native tcolorbox solution. The content of a tcolorbox is processed with the typical settings of parbox and minipage. You can switch this kind processing by setting parbox=false to do a mimicry of 'normal' processing.

You find this in the tcolorbox documentation as Text Characteristics (4.18 Text Characteristics on p. 98, for current tcolorbox version 4.22 [2019/11/15]).

\documentclass{article}

\usepackage{tcolorbox}
\usepackage{lipsum}
\usepackage{parskip}

\begin{document}

\begin{tcolorbox}[parbox=false]
\lipsum[1]

\lipsum[2]
\end{tcolorbox}

\lipsum
\end{document}

One option would be to use the settings from parskip inside a new environment defined using tcolorbox:

\documentclass{article}
\usepackage{tcolorbox}
\usepackage{lipsum}
\usepackage{parskip}

\newenvironment{mycolorbox}[1][]
  {\if\detokenize{#1}\relax\relax
      \begin{tcolorbox}
    \else
      \begin{tcolorbox}[#1]
    \fi
  \parskip=0.5\baselineskip \advance\parskip by 0pt plus 2pt
  \parindent=0pt
}
  {\end{tcolorbox}}

\begin{document}

\begin{mycolorbox}
\lipsum[1]

\lipsum[2]
\end{mycolorbox}

\lipsum

\end{document}