Here follows a minimal working example:
\documentclass[a4paper,11pt,twoside,openright]{memoir}
\setlrmargins{*}{*}{1}
\setulmargins{*}{*}{1}
\checkandfixthelayout[nearest]
\usepackage{color}
\usepackage{xcolor}
\usepackage{titletoc}
\settocdepth{subsection}
\setsecnumdepth{subsection}
\makeatletter
\def\@endpart{
%
\vfil
\textcolor{red}{\hrule height 1.5 pt}
\setcounter{tocdepth}{0}
\startcontents[parts]
\printcontents[parts]{}{0}{}
\textcolor{red}{\hrule height 1.5 pt}
\par\bigskip
\vfil\newpage
\if@twoside
\if@openright
\null
\thispagestyle{empty}%
\newpage
\fi
\fi
\if@tempswa
\twocolumn
\fi}
\makeatother
\begin{document}
\part{First Part}
\chapter{First Chapter}
\section{First Section}
\subsection{First Subsection}
\chapter{Second Chapter}
\section{Second Section}
\subsection{Second Subsection}
\chapter{Third Chapter}
\section{Third Section}
\subsection{Third Subsection}
\end{document}
And this is how the first page looks like:

If you are interested in tweaking it a little bit more, check out this post: Pretty List of Figures and List of Tables.
Inspired by the work of Gonzalo Medina, I have managed to do a quite nice Part TOC which you can use. Check out the link and reuse some of the code if you find it useful.
By the way, the choice of the colours were made so that I could easily find the code blocks in the preamble, coding for the different elements in the TOC and Part TOC.
Pictures can be seen below:

