If I understand you correctly, you want to start the \part on odd pages (not left-sided, is it?) and keep \chapter to start on both even and odd pages.
You can do this by using the \cleardoublepage command which does the same thing as \clearpage but also adds a blank page if needed. You can then redefine the \part macro to call \cleardoulepage before executing the rest of its content.
\usepackage{etoolbox}
\pretocmd{\part}{\cleardoublepage}{}{}
For more information on the \pretocmd macro, please refer to egreg's excellent post.
MWE
Here's a minimal working example:
\documentclass[twoside, openany]{memoir}
\usepackage{lipsum}
\usepackage{etoolbox}
\pretocmd{\part}{\cleardoublepage}{}{}
\begin{document}
\part{The beginning of a new era}
\chapter{The evolution}
\lipsum[1-6]
\part{Another era}
\chapter{The second evolution}
\lipsum[1-6]
\end{document}
Note the use of the openany option which opens any page for starting new chapters, as by default, a chapter will always start on odd pages.