Compared to the class book, chapters are reset at the begin of a new part and figures and tables are reset at the start of a new section. There are several methods to do that, see the links in the comment of Count Zero above.
It remains the problem, that the numbering is not unique and it is not clear, to which part figure 2.1.1 belong to, for example. The following example adds the part, if the reference points outside the current part:

Each reference is prefixed by \p@<counter> that is redefined to expand to \RefPart with the current part number. \RefPart is defined using \DeclareRobustCommand, because it must not be expanded, when the label is written to the .aux file. This way the \RefPart is executed at time of \ref that can be called in both parts generating different references (with and without part prefix).
Additionally the example is tested with hyperref.
\documentclass{book}
\usepackage{hyperref}
\usepackage{bookmark}
\makeatletter
% numbering of chapters inside part:
\@addtoreset{chapter}{part}
\@ifpackageloaded{hyperref}{%
\renewcommand*{\theHchapter}{%
\theHpart.\arabic{chapter}%
}%
}{}
% numbering of figure/table inside section:
\@addtoreset{figure}{section}
\renewcommand*{\thefigure}{\thesection.\arabic{figure}}
\@addtoreset{table}{section}
\renewcommand*{\thetable}{\thesection.\arabic{table}}
\DeclareRobustCommand*{\RefPart}[1]{%
\begingroup
\def\x{#1}%
\protected@edef\y{\thepart}%
\ifx\x\y
\else
\x.%
\fi
\endgroup
}
\renewcommand*{\p@table}{\RefPart{\thepart}}
\renewcommand*{\p@figure}{\RefPart{\thepart}}
\makeatother
\begin{document}
\tableofcontents
\part{First part}
\chapter{Chapter A}
Reference inside part: \ref{figE}\\
reference to other part: \ref{figEE}
\chapter{Chapter B}
\section{Section C}
\subsection{Subsection D}
\begin{figure}
\caption{Figure E}
\label{figE}
\end{figure}
\part{Second part}
\chapter{Chapter AA}
Reference to other part: \ref{figE}\\
reference inside part: \ref{figEE}
\chapter{Chapter BB}
\section{Section CC}
\subsection{Subsection DD}
\begin{figure}
\caption{Figure EE}
\label{figEE}
\end{figure}
\end{document}