\documentclass[openany,]{scrbook}
\usepackage[english]{babel}
\usepackage{blindtext}

\usepackage{etoolbox}

\makeatletter
% First, modify the \@endpart macro.
\def\@endpart{}

% Next, copy the \chapter macro to \nonewpagechapter, and ...
% ... suppress page-breaking instructions in the modified macro
\let\nonewpagechapter\chapter
\patchcmd\nonewpagechapter{\if@openright\cleardoublepage\else\clearpage\fi}{}{}{}

% Third, suppress vertical whitespace before "Part xx" material
\patchcmd{\part}{\null\vfil}{}{}{}

\makeatother

\begin{document}
\part{foo}
\nonewpagechapter{Uno} % starts on same page as "Part I ... foo" header
\blindtext
\chapter{Dos}          % starts on a new page
\part{bar}             % starts on a new page
\nonewpagechapter{Tres}% starts on same page as "Part II ... bar" header
\blindtext
\end{document}