\documentclass{article}
\usepackage{etoolbox}
\usepackage{threeparttable}

\makeatletter
\apptocmd{\TPT@endtabbox}
  {\AddToHookNext{env/\@currenvir/after}{\IgnorePar}}
  {}{\ERROR}
\makeatother

\AtBeginDocument{%
  \NewLogicalBlock{threeparttable}
  \SetTag{threeparttable}{div}
  \SetBlockProperty{threeparttable}{class}{threeparttable}

  \NewLogicalBlock{tablenotes}
  \SetTag{tablenotes}{div}
  \SetBlockProperty{tablenotes}{class}{tablenotes}
}

\begin{document}

\begin{threeparttable}
  \begin{tabular}{l}
    A
  \end{tabular}
  %
  \begin{tablenotes}
    \item[a] Some note
  \end{tablenotes}
  %
  \caption{A fascinating table}
\end{threeparttable}

\end{document}
