To achieve your objective without loading a package such as titlesec, one must (i) modify the command \@endpart and (ii) change the \chapter command because it, too, inserts a page break by default. Although you didn't specify this explicitly, I further assume that you (iii) want to have the "Part xx" material placed at the top of the page rather than in the middle. By the way, I assume that your document is set in "onecolumn" rather than in "twocolumn" mode. Taking care of the twocolumn case isn't all that more difficult, but it does add some overhead which I gather isn't required here.
The following code shows how to do this for the case of the report document class. Note that I've created a command called \nonewpagechapter and have left the standard \chapter command untouched. That way, you can still use the \chapter command for all those occasions when you do want the chapter to start on a new page. Of course, if \nonewpagechapter is too wordy for your taste, you can give it a more succinct name, such as \nchapter...
\documentclass{report}
\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
\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
\end{document}
Addendum If someone were using the scrbook (Koma-Script book) document class instead of the report document class, he/she would have to replace the command
\patchcmd{\part}{\null\vfil}{}{}{}
with
\patchcmd{\part}{\partheadstartvskip}{}{}{}
in order to suppress the insertion of extra whitespace above the "Part xx" line. The other patch commands could be used without modification.
chapterproduces also a page break. However you can use a package liketitlesec. – Marco Daniel Jan 27 '12 at 19:56