So, similar to the last answer, you will need to pull in the lines appropriate to paragraph and subparagraph. These use some different things to make the headings, including different ways of spacing and indenting the sections. I've annotated a few things in the MWE below. See also this great answer for more about how the \@startsection command works so you can change some of these parameters yourself.
I have not included your previous solution (adding chapters) for clarity of this solution. The two can co-exist very nicely.
The MWE now becomes:
\documentclass[boldsubject,italicsignature,smartindent]{letteracdp}
\usepackage[italian]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{layaureo}
\usepackage{lipsum}
\usepackage{blindtext} % Just for demo
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
\makeatletter
%%% Add paragraphs and subparagraphs
\newcounter {paragraph}
\newcounter {subparagraph}[paragraph]
\newcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
{3.25ex \@plus1ex \@minus.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
%% The subparagraph indents by \parindent, which is set to 72.12128pt or roughly an inch
%% Redefine it in the letter body (after \opening) if you would like it smaller
\newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{3.25ex \@plus1ex \@minus .2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
\makeatother
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
\begin{document}
\begin{letter}[John Doe]{
Prof.~John \textsc{Doe}\\
Department of Electrical Engineering\\
Stanford University\\
Stanford, CA~94305 (U.S.A.)
}[
This is the optional subject of the letter.
]
\opening{Dear Prof.~Doe,}
% Change the paragraph indent here if desired
\setlength\parindent{1cm}
\blindtext
\paragraph{This is a paragraph} \blindtext
\subparagraph{This is a subparagraph} \blindtext
\closing{Sincerely,}
\end{letter}
\end{document}
This presents (in part):
