The immediate problem is that you need the counter for paragraphs to be reset when you start a new subsubsubsection. To do this, all you need to do is add:
\@addtoreset{paragraph}{subsubsubsection}
Note, though, that you probably will also find you want to add some further resets depending on the structure of your document.
More generally, I have to say that the fact that you find the need to do this indicates to me that you need to rethink the structure of your document. Even if you are writing a book or thesis (for which article
would not be a suitable class), 9 levels is just going to be confusing. The only exception is probably legal documentation which seems to positively require insane levels of hierarchy. I think you should really be asking what you expect your readers to gain from the structure in terms of clarity. If the answer is nothing or, worse, greater complexity and confusion, you know you need to rethink.
However, this is TeX SE and so I can provide a 'solution' with a clean conscience since the site is not in the business of ensuring people write well, but only of assisting them to write whatever they do in TeX.
\documentclass{article}
\usepackage{titlesec}
\titleclass{\subsubsubsection}{straight}[\subsection]
\newcounter{subsubsubsection}[subsubsection]
\renewcommand\thesubsubsubsection{\thesubsubsection.\arabic{subsubsubsection}}
\renewcommand\theparagraph{\thesubsubsubsection.\arabic{paragraph}}
\renewcommand\thesubparagraph{\theparagraph.\arabic{subparagraph}}
\titleformat{\subsubsubsection}
{\normalfont\normalsize\bfseries}{\thesubsubsubsection}{1em}{}
\titlespacing*{\subsubsubsection}
{0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}
\makeatletter
\renewcommand\paragraph{\@startsection{paragraph}{5}{\z@}%
{3.25ex \@plus1ex \@minus.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
\renewcommand\subparagraph{\@startsection{subparagraph}{6}{\parindent}
{3.25ex \@plus1ex \@minus .2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
\def\toclevel@subsubsubsection{4}
\def\toclevel@paragraph{5}
\def\toclevel@paragraph{6}
\def\l@subsubsubsection{\@dottedtocline{4}{7em}{4em}}
\def\l@paragraph{\@dottedtocline{5}{10em}{5em}}
\def\l@subparagraph{\@dottedtocline{6}{14em}{6em}}
\@addtoreset{subsubsubsection}{section}
\@addtoreset{subsubsubsection}{subsection}
\@addtoreset{paragraph}{subsubsubsection}
\makeatother
\setcounter{secnumdepth}{6}
\setcounter{tocdepth}{6}
\author{Author}
\title{Title}
\date{2014}
\begin{document}
\tableofcontents
\section{Section 1}
\subsection{subsection 1}
\subsubsection{subsubsection 1}
\subsubsubsection{Subsubsubsection 1}
Some description here
\paragraph{paragraph 1}
Some description here
\paragraph{Paragraph 2}
Some description here
\subsubsubsection{subsubsubsection 2}
Description here
\subsubsubsection{subsubsubsection 3}
description here
\subsubsubsection{subsubsubsection 4}
description here
\paragraph{paragraph 1}
Some description here
\paragraph{paragraph 2}
some description here
\paragraph{Paragraph 3}
\subsubsubsection{subsubsubsection 5}
description here
\end{document}

Do you just want something like this? I'm still not entirely clear what the command should do about the page but maybe you don't want it to do anything?
\documentclass{article}
\usepackage{titlesec}
\titleformat{\section}[runin]{\Large\bfseries}{\thesection}{0pt}{}[]
\setcounter{secnumdepth}{0}
\begin{document}
\section{Name of the section} (page)
\paragraph{Name of the section} (page)
\titleformat*{\paragraph}{\Large \bfseries}
\paragraph{Name of the section} (page)
\end{document}

Best Answer
As suggested by egreg in the comments of the question, the solution was as simple as updating the packages. I didn't know that the texlive package available in the Ubuntu repository was not the most up to date. I followed the instructions on the link below to get the most updated version of texlive and now it works.
http://tug.org/texlive/quickinstall.html